Make sure… The Six Trigonometric Functions and Special Triangles To determine the trigonometric function values of an acute angle can be tricky especially if the angle is not a special one. Which angle between \(0^\circ \) and \(360^\circ \) has the same terminal side (and hence the same trigonometric function values) as \(\theta\,\)? Question 1 : Determine the exact values of sin θ, cos θ and tan θ if the terminal arm of an angle in standard position passes through the given point. It means that the relationship between the angles and sides of a triangle are given by these trig functions. Solution for Calculate the six trigonometric function values of an angle e in standard position whose terminal side goes through the point (-12, 16). Thus, \(\sin\;\theta = \frac{y}{r} = \frac{-3}{5} \) and \(\tan\;\theta = \frac{y}{x} = \frac{-3}{-4} = \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 1.4: Trigonometric Functions of Any Angle, [ "article:topic", "authorname:mcorral", "showtoc:no", "transcluded:yes", "license:gnufdl", "source-math-3247" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FCourses%2FRiverside_City_College%2FRandom_Course%2F1%253A_Right_Triangle_Trigonometry_Angles%2F1.4%253A_Trigonometric_Functions_of_Any_Angle, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 1.3: Applications and Solving Right Triangles, GNU Free Documentation License, Version 1.2, information contact us at info@libretexts.org, status page at https://status.libretexts.org. Finding Trig Functions if the Terminal Side Passes Through given Point - Examples. Find the exact value of each of the six trigonometric functions of θ. \cos\;\theta ~=~ \dfrac{x}{r} \qquad\qquad 3x+y=0, x ≤ 0 I have no effing idea on how I would determine the right point on the graph to select to use to solve this problem. (a) Since \(928^\circ = 2 \times 360^\circ + 208^\circ \), then \(\theta \) has the same terminal side as \(208^\circ \), as in Figure 1.4.7. \tan\;270^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{-1}{0} \;=\; \text{undefined}\], \[\nonumber \csc\;270^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{1}{-1} \;=\; -1\qquad Lastly, for \(270^\circ \) use the point \((0,-1) \) so that \(r = 1 \), \(x = 0 \), and \(y = -1 \). An angle is in standard position in the coordinate plane if its vertex is located at the origin and one ray is on the positive x-axis. In the lesson you learned that the terminal side of the angle intersects the unit circle at the point . Trigonometric functions are also known as a Circular Functions can be simply defined as the functions of an angle of a triangle. Now let \(\theta \) be any angle. \sec\;330^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{2}{\sqrt{3}} \qquad \frac{3}{4} \). The signs of the sine and cosine are determined from the x– and y-values in the quadrant of the original angle. Likewise, for \(180^\circ \) use the point \((-1,0) \) so that \(r = 1 \), \(x = -1 \), and \(y = 0 \). When \(\theta \) is in QIII, we see from Figure 1.4.8(b) that the point \((-4,-3) \) is on the terminal side of \(\theta \), and so we have \(x = -4 \), \(y = -3 \), and \(r = 5 \). In this position, the vertex (B) of the angle is on the origin, with a fixed side lying at 3 o'clock along the positive x axis. Since reciprocals have the same sign, \(\csc\;\theta \) and \(\sin\;\theta \) have the same sign, \(\sec\;\theta \) and \(\cos\;\theta \) have the same sign, and \(\cot\;\theta \) and \(\tan\;\theta \) have the same sign. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. Keep in mind the sign of the functions during these conversions to the reference angle. Legal. Consider the right triangle on the left.For each angle P or Q, there are six functions, each function is the ratio of two sides of the triangle.The only difference between the six functions is which pair of sides we use.In the following table 1. a is the length of the side adjacent to the angle (x) in question. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. See explanation. Solution for Find the values of the six trigonometric functions of 0 if the terminal side of an angle 0 in standard position contains the point (7, -1). How would I "Find the six trigonometric functions for the angle theta whose terminal side passes through the point (-8,-5)"?. Naturally, the values of all six trig functions of an angle are the same as those angles with which it is co-terminal. We then define the trigonometric functions of \(\theta \) as follows: \[\label{1.2} \sin\;\theta ~=~ \dfrac{y}{r} \qquad\qquad P(4, -7) Since \(360^\circ \) represents one full revolution, the trigonometric function values repeat every \(360^\circ \). Definitions of the Six Trigonometric Functions: General Case Let θ be an angle drawn in standard position, and let ( , )Px y represent the point where the terminal side of the angle intersects the circle x22 2+yr= . Trig calculator finding sin, cos, tan, cot, sec, csc. Angles such as these, which have the same initial and terminal sides, are called coterminal. Hence: \[\nonumber \sin\;330^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{-1}{2} \qquad \cos\;330^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{\sqrt{3}}{2} \qquad Pick any point \((x,y) \) on the terminal side of \(\theta \) a distance \(r>0 \) from the origin (see Figure 1.4.3(c)). ... Three trigonometric functions for a given angle are shown below. What is the reference angle for \(\theta\,\)? \tan\;330^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{-1}{\sqrt{3}}\], \[\nonumber \csc\;330^\circ \;=\; \dfrac{r}{y} \;=\; -2 \qquad Drawing that triangle in QIV so that the hypotenuse is on the terminal side of \(330^\circ \) makes \(r = 2 \), \(x=\sqrt{3} \), and \(y=-1 \). Find the exact values of the five remaining trigonometric functions of θ. 2. o is the length of the side opposite the angle. Find the values of all six trig functions. \tan\;180^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{0}{-1} \;=\; 0\], \[\nonumber \csc\;180^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\quad\;\;\; Example 1 : (5/7, 2√6/7) is a point on the terminal side of an angle θ in standard position. Then draw a negative angle in Drawing that triangle in QIII so that the hypotenuse is on the terminal side of \(225^\circ \) makes \(r = \sqrt{2} \), \(x=-1 \), and \(y=-1 \). By Example 1.7 in Section 1.2, we see that we can use the point \((-1,\sqrt{3})\) on the terminal side of the angle \(120^\circ\) in QII, since we saw in that example that a basic right triangle with a \(60^\circ\) angle has adjacent side of length \(1\), opposite side of length \(\sqrt{3}\), and hypotenuse of length \(2\), as in the figure on the right. Sine = 3/5 = 0.6 Cosine = 4/5 = 0.8 Tangent =3/4 = .75 Cotangent =4/3 = 1.33 Secant =5/4 = 1.25 Cosecant =5/3 = 1.67 Begin by drawing the terminal side in standard position and drawing the associated triangle. That acute angle has a special name: if \(\theta \) is a nonacute angle then we say that the reference angle for \(\theta \) is the acute angle formed by the terminal side of \(\theta \) and either the positive or negative \(x\)-axis. \tan\;225^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{-1}{-1} \,=\, 1\], \[\nonumber \csc\;225^\circ \;=\; \dfrac{r}{y} \;=\; -\sqrt{2} \qquad (2, 6) is a point on the terminal side of . Hence: \[\nonumber \sin\;120^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{\sqrt{3}}{2} \qquad The terminal side of an angle in standard position passes through P(-3, -4). Again, you could think of the line segment from the origin to \((0,1) \) as a degenerate right triangle whose base has length \(0 \) and whose height equals the length of the hypotenuse. \cot\;180^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{-1}{0} \;=\; \text{undefined}\]. So in Example 1.20, we see that \(60^\circ \) is the reference angle for the nonacute angle \(\theta = 120^\circ\); in Example 1.21, \(45^\circ \) is the reference angle for \(\theta = 225^\circ\); and in Example 1.22, \(30^\circ \) is the reference angle for \(\theta = 330^\circ \). Question 1159215: A point on the terminal side of angle θ is given. Underneath the calculator, six most popular trig functions will appear - three basic ones: sine, cosine and tangent, and their reciprocals: cosecant, secant and cotangent. For example, \(\sin\;360^\circ = \sin\;0^\circ \), \(\cos\;390^\circ = \cos\;30^\circ \), \(\tan\;540^\circ = \tan\;180^\circ \), \(\sin\;(-45^\circ) = \sin\;315^\circ \), etc. (-6,-5) Answer by Theo(11212) (Show Source): \tan\;0^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{0}{1} \;=\; 0\], \[\nonumber \csc\;0^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\qquad This trigonometry video tutorial explains how to evaluate trigonometric functions given a point on the terminal side. Find the exact values of the five remaining trigonometric functions of θ. Hence: \[\nonumber \sin\;225^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{-1}{\sqrt{2}} \qquad Ex. Evaluate the six trigonometric functions for theta. In this position, the vertex (B) of the angle is on the origin, with a fixed side lying at 3 o'clock along the positive x axis. \sec\;0^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{1}{1} \;=\; 1 \qquad Let \(\theta = 928^\circ \). The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle. In the figure below, drag point A and see how the position of the terminal side BA defines the angle. \tan\;90^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\], \[\nonumber \csc\;90^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{1}{1} \;=\; 1\qquad In Examples 1.20-1.22, we saw how the values of trigonometric functions of an angle \(\theta \) larger than \(90^\circ \) were found by using a certain acute angle as part of a right triangle. Adopted a LibreTexts for your class? ; An angle’s reference angle is the size angle, [latex]t[/latex], formed by the terminal side of the angle [latex]t[/latex] and the horizontal axis. The basic trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant. sine of theta = -77/85 cosine of theta = 36/85 tangent of theta = -77/36. }\)) Show that the point \(P^{\prime}(24, 10)\) also lies on the terminal side of the angle. You could think of the line segment from the origin to the point \((1,0) \) as sort of a degenerate right triangle whose height is \(0 \) and whose hypotenuse and base have the same length \(1 \). \sec\;180^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{1}{-1} \;=\; -1\quad\;\;\; As in the acute case, by the use of similar triangles these definitions are well-defined (i.e. Regardless, in the formulas we would use \(r = 1 \), \(x = 1 \), and \(y = 0 \). Notice that in the case of an acute angle these definitions are equivalent to our earlier definitions in terms of right triangles: draw a right triangle with angle \(\theta \) such that \(x = \text{adjacent side} \), \(y = \text{opposite side} \), and \(r = \text{hypotenuse} \). In trigonometry an angle is usually drawn in what is called the "standard position" as shown above. (b) \(928^\circ \) and \(208^\circ \) have the same terminal side in QIII, so the reference angle for \(\theta = 928^\circ \) is \(208^\circ - 180^\circ = 28^\circ \). There are some special angles, however, that can readily give us … \cot\;270^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{0}{-1} \;=\; 0\]. Have questions or comments? Click here to let us know! Thus, \(\sin\;\theta = \frac{y}{r} = \frac{3}{5} \) and \(\tan\;\theta = \frac{y}{x} = \frac{3}{-4} \). We know that \(225^\circ = 180^\circ + 45^\circ \). In general, if two angles differ by an integer multiple of \(360^\circ\) then each trigonometric function will have equal values at both angles. By Example 1.6 in Section 1.2, we see that we can use the point \((-1,-1) \) on the terminal side of the angle \(225^\circ \) in QIII, since we saw in that example that a basic right triangle with a \(45^\circ \) angle has adjacent side of length \(1 \), opposite side of length \(1 \), and hypotenuse of length \(\sqrt{2} \), as in the figure on the right. Hence: \[\nonumber \sin\;180^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{0}{1} \;=\; 0 \qquad \cos\;90^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{0}{1} \;=\; 0 \qquad Find the exact trigonometric function values of any angle whose reference angle measures 30°, 45°, or 60°. We know that \(330^\circ = 360^\circ - 30^\circ \). r = √x2 + y2 Ex. Example 3: Draw a 120o angle in standard position. \cos\;270^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{0}{1} \;=\; 0 \qquad \cot\;90^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{0}{1} \;=\; 0\]. Determine the trigonometric function values of angle θ. \cot\;\theta ~=~ \dfrac{x}{y}\]. Drawing that triangle in QII so that the hypotenuse is on the terminal side of \(120^\circ\) makes \(r = 2\), \(x=-1\), and \(y=\sqrt{3}\). We have \(r = 1 \), \(x = 0 \), and \(y = 1 \), and hence: \[\nonumber \sin\;90^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{1}{1} \;=\; 1 \qquad 16. Chapter 6: Trigonometric Functions Expand/collapse global location 6.2: Angles Last updated; Save as PDF Page ID ... draw the initial side along the positive \(x\)-axis. What are the coordinates of point (x, y) on the terminal ray of angle theta, assuming that the values above were not simplified? I don't even know where to start. So unlike the previous examples, we do not have any right triangles to draw. The ray on the x-axis is called the initial side and the other ray is called the terminal side. The equation, with a restriction of x, is the terminal side of an angle θ in standard position. Figure 1.4.7. The terminal side of an angle passes through the point P = (10,3). Please please help Find the value of all six trigonometric functions evaluated at e. cos sin @ tane csc 0 = sec coto \cot\;0^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\]. Hence: \[\nonumber \sin\;0^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{0}{1} \;=\; 0 \qquad I learned this material over 2 years ago and since then have forgotten. The following table summarizes the values of the trigonometric functions of angles between \(0^\circ \) and \(360^\circ \) which are integer multiples of \(30^\circ \) or \(45^\circ\): Table 1.3 Table of trigonometric function values. Why?) Find the six trig values of (4, 3) if it's a point on the terminal side of an angle in standard position. From the Pythagorean theorem, the hypotenuse can be found. \cot\;120^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{-1}{\sqrt{3}}\]. \sec\;225^\circ \;=\; \dfrac{r}{x} \;=\; -\sqrt{2} \qquad Terminal side of an angle - trigonometry. For instance, for the angle \(0^\circ \) use the point \((1,0) \) on its terminal side (the positive \(x\)-axis), as in Figure 1.4.6. \cos\;225^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{-1}{\sqrt{2}} \qquad See also Angle definition and properties - Trigonometry, Angle definition and properties - Trigonometry, Tangent function (tan) in right triangles, Cotangent function cot (in right triangles), Cosecant function csc (in right triangles), Finding slant distance along a slope or ramp. Hence: \[\nonumber \sin\;270^\circ \;=\; \dfrac{y}{r} \;=\; \dfrac{-1}{1} \;=\; -1 \qquad \sec\;\theta ~=~ \dfrac{r}{x} \qquad\qquad Similarly, from Figure 1.4.6 we see that for \(90^\circ \) the terminal side is the positive \(y\)-axis, so use the point \((0,1) \). In most cases, the values can’t be obtained without the use of special table of values, calculator or any computing device. \sec\;270^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\qquad So it suffices to remember the signs of \(\sin\;\theta \), \(\cos\;\theta \), and \(\tan\;\theta\): For an angle \(θ\) in standard position and a point \((x, y)\) on its terminal side: Michael Corral (Schoolcraft College). (Note that \(r = \sqrt{ x^2 + y^2 } \). Answer to: Find the exact values of each of the six trigonometric functions at the point (-1, -3) on the terminal side of an angle theta. For instance, for the angle \(0^\circ \) use the point \((1,0) \) on its terminal side (the positive \(x\)-axis), as in Figure 1.4.6. \cos\;180^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{-1}{1} \;=\; -1 \qquad \tan\;\theta ~=~ \dfrac{y}{x}\], \[\label{1.3}\csc\;\theta ~=~ \dfrac{r}{y} \qquad\qquad they do not depend on which point \((x,y) \) we choose on the terminal side of \(\theta\)). Thus, either \(\fbox{\(\sin\;\theta = \frac{3}{5} \) and \(\tan\;\theta = -\frac{3}{4}\)}\) or \(\fbox{\(\sin\;\theta = -\frac{3}{5} \) and \(\tan\;\theta = \frac{3}{4}\)}\). However, the values of the trigonometric functions are easy to calculate by picking the simplest points on their terminal sides and then using the definitions in formulas Equation \ref{1.2} and Equation \ref{1.3}. By Example 1.7 in Section 1.2, we see that we can use the point \((\sqrt{3},-1) \) on the terminal side of the angle \(225^\circ \) in QIV, since we saw in that example that a basic right triangle with a \(30^\circ \) angle has adjacent side of length \(\sqrt{3} \), opposite side of length \(1 \), and hypotenuse of length \(2 \), as in the figure on the right. \cos\;0^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{1}{1} \;=\; 1 \qquad a) P (-5, 12) We do that by dividing the angle measure in degrees by 360°. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Using the Pythagorean Theorem calculate the missing side the hypotenuse. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. These angles are different from the angles we have considered so far, in that the terminal sides lie along either the \(x\)-axis or the \(y\)-axis. Trigonometry-basics/1128094 (2018-11-05 16:42:40): A point P is on the terminal side of angle theta. \sec\;120^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{2}{-1} \;=\; -2 \qquad \cot\;225^\circ \;=\; \dfrac{x}{y} \;=\; \dfrac{-1}{-1} \,=\, 1\]. \(\sin\;\theta \) has the same sign as \(y\), \(\cos\;\theta \) has the same sign as \(x\), \(\tan\;\theta \) is positive when \(x \) and \(y \) have the same sign, \(\tan\;\theta \) is negative when \(x \) and \(y \) have opposite signs. terminal side. terminal side of an angle in standard position. However, the values of the trigonometric functions are easy to calculate by picking the simplest points on their terminal sides and then using the definitions in formulas Equation \ref{1.2} and Equation \ref{1.3}. Find the six trig values of (­2, ­1) if it's a point on the terminal side of an angle in standard position.} Also, notice that \(| \sin\;\theta | \le 1 \) and \( | \cos\;\theta | \le 1 \), since \(| y | \le r \) and \( | x | \le r \) in the above definitions. For example, this would give us \(\sin\;\theta = \frac{y}{r} = \frac{\text{opposite}}{\text{hypotenuse}} \) and \(\cos\;\theta = \frac{x}{r} = \frac{\text{adjacent}}{\text{hypotenuse}} \), just as before (see Figure 1.4.4(a)). We say that \(\theta \) is in standard position if its initial side is the positive \(x\)-axis and its vertex is the origin \((0,0) \). Find the equation of the terminal side of the angle in the previous example. \cot\;330^\circ \;=\; \dfrac{x}{y} \;=\; -\sqrt{3}\]. Letting the positive x-axis be the initial side of an angle, you can use the coordinates of the point where the terminal side intersects with the […] Suppose θ is an angle in the standard position whose terminal side is in Quadrant IV and cotθ = -6/7 . Co-terminal angles have a common terminal side. The terminal side of the angle intersects the unit circle at (0, -1). For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. One way to find the values of the trig functions for angles is to use the coordinates of points on a circle that has its center at the origin. \tan\;120^\circ \;=\; \dfrac{y}{x} \;=\; \dfrac{\sqrt{3}}{-1} \,=\, -\sqrt{3}\], \[\nonumber \csc\;120^\circ \;=\; \dfrac{r}{y} \;=\; \dfrac{2}{\sqrt{3}} \qquad In trigonometry an angle is usually drawn in what is called the "standard position" as shown above. When \(\theta \) is in QII, we see from Figure 1.4.8(a) that the point \((-4,3) \) is on the terminal side of \(\theta \), and so we have \(x = -4 \), \(y = 3 \), and \(r = 5 \). If the angle is in quadrant III, rotate 180°. We know \(120^\circ = 180^\circ - 60^\circ\). Use this ordered pair to find the six trig functions of . Note that \(\csc\;0^\circ \) and \(\cot\;0^\circ \) are undefined, since division by \(0 \) is not allowed. Key Concepts. (Hint: The terminal side lies on a line that goes through the origin and the point \((12,5)\text{. If the point is given on the terminal side of an angle, then: Calculate the distance between the point given and the origin: r=sqrt(x^2+y^2) Here it is: r=sqrt(7^2+24^2)=sqrt(49+576)=sqrt(625)=25 Now we can calculate all 6 trig, functions: sinalpha=y/r=24/25 cosalpha=x/r=7/25 tanalpha=y/x=24/7=1 3/7 cotalpha=x/y=7/24 … I don't even know how to graph 3x+y=0, x ≤ 0 to even think of what to select. The sign of a trigonometric function is dependent on the signs of the coordinates of the points on the terminal side of the angle. Example 1: Find the six trigonometric functions of an angle α that is in standard position and whose terminal side passes through the point (−5, 12). Problem : Which trigonometric functions are independent of the distance between a point and the origin (when the terminal side of an angle in standard position contains that point)? The other side, called the 'terminal side' is the one that can be anywhere and defines the angle. By knowing in which quadrant the terminal side of an angle lies, you also know the signs of all the trigonometric functions. To find the trigonometric functions of an angle, enter the chosen angle in degrees or radians. Find the exact values of the six trigonometric functions. \cos\;120^\circ \;=\; \dfrac{x}{r} \;=\; \dfrac{-1}{2} \qquad \sec\;90^\circ \;=\; \dfrac{r}{x} \;=\; \dfrac{1}{0} \;=\; \text{undefined}\qquad Here we are going to see finding trig functions if the terminal side passes through given point. The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2. Three trigonometric functions for a given angle are shown below.