find other five trigonometric functions

Find the values of the other five trigonometric functions in each of the following: cos in quadrant II. We can find the sine using the Pythagorean Identity, cos 2 t + sin 2 t = 1 , and the remaining functions by relating them to sine and cosine. The graph is not symmetrical about the y-axis. Have questions or comments? The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x– and y-values in the original quadrant. If If x and x are in the second quadrant, find the other five trigonometric functions. By showing that \(\frac{ \sec t}{ \tan t}\) can be simplified to \( \csc t\),we have, in fact, established a new identity. \( \sec t= \sqrt{2},\csc t=\sqrt{2}, \tan t=1, \cot t=1\). ... Ch. We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. It;s crucial to be familiar with the Pythagorean trigonometric identity and trigonometric identities corresponding to relations of sine and cosine with other trigonometric functions. Use properties of … Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions. The values of trigonometric functions of special angles can be found by mathematical analysis. Since x is in lllrd Quadrant sin and cos will be negative But tan will be positive Given cos x = (−1)/2 We know that sin2 x + cos2 x = 1 sin2 x + ( (−1)/2)^2 = 1 sin2 x + 1/4 = 1 sin2 x = 1 – 1/4 sin2 x = (4 − 1)/4 sin2 x = (4 −1)/4 sin2x = 3/4 sin x = ±√ (3/4) sin x = ± √3/2 Since x is … 4:12 000+ LIKES. Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function. Adopted a LibreTexts for your class? Now consider the function \(f(x)=x^3\), shown in Figure \(\PageIndex{6}\). The result is another function that indicates its rate of change (slope) at a particular values of x. tan θ = -2/1. We can test each of the six trigonometric functions in this fashion. 6 - Find the values of the six trigonometric functions... Ch. Find \( \sin t, \cos t, \tan t, \sec t, \csc t,\) and \( \cot t\). See, Trigonometric functions of angles outside the first quadrant can be determined using reference angles. sec (theta) = hyp/adj = sqrt (5)/1 = sqrt (5) csc (theta) = hyp/opp = sqrt (5)/2. If the secant of angle t is 2, what is the secant of \(−t\)? A² = 5. A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is \(\frac{1}{12}\) or less, regardless of its length. A function is said to be even if \(f(−x)=f(x)\) and odd if \(f(−x)=−f(x)\). \\ \text{} & = \csc t \end{array}\]. The angle between this angle’s terminal side and the x-axis is \(\frac{π}{6}\), so that is the reference angle. Find the values of the six trigonometric functions of angle \(t\) based on Figure \(\PageIndex{9}\) . For more on this see Derivatives of trigonometric functions. Answer by Edwin McCravy (18456) ( Show Source ): You can put this solution on YOUR website! For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Buy Find launch. See, Trigonometric functions can also be found from an angle. In six trigonometric ratios sin, cos, tan, csc, sec and cot, if the value of one of the ratios is given, we can find the values of the other five functions. \[ \begin{align} \cos (−\dfrac{5π}{6}) &=−\dfrac{\sqrt{3}}{2}, \sin (−\dfrac{5π}{6})=−\dfrac{1}{2}, \tan (−\dfrac{5π}{6}) = \dfrac{\sqrt{3}}{3} \\ \sec (−\dfrac{5π}{6}) &=−\dfrac{2\sqrt{3}}{3}, \csc (−\dfrac{5π}{6})=−2, \cot (−\dfrac{5π}{6})=\sqrt{3} \end{align} \]. Ex 3.2, 4 Find the values of other five trigonometric functions if sec x = 13/5 , lies in fourth quadrant. We can derive some useful identities from the six trigonometric functions. To define the remaining functions, we will once again draw a unit circle with a point \((x,y)\) corresponding to an angle of \(t\),as shown in Figure \(\PageIndex{1}\). \[\begin{array}{lll} \dfrac{\sec t}{\tan t} & =\dfrac{1 / \cos t}{ \sin t / \cos t} & \text{To divide the functions, we multiply by the reciprocal.} tan θ = 3. See. Replace the known values in the equation . Access these online resources for additional instruction and practice with other trigonometric functions. With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Persian and Arab mathematicians, including the cosine, tangent, cotangent, secant and cosecant. \[ \begin{align*} \tan \dfrac{π}{6} & = \dfrac{ \sin \frac{π}{6}}{\cos \frac{π}{6}} \\ & = \dfrac{\frac{1}{2} }{\frac{\sqrt{3}}{2}}=\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3} \\ \sec \dfrac{π}{6} &= \dfrac{1}{ \cos \frac{π}{6}} \\ & = \dfrac{1}{\frac{\sqrt{3}}{2}} = \dfrac{2}{\sqrt{3}}= \dfrac{2\sqrt{3}}{3} \\ \csc \dfrac{π}{6} &= \dfrac{1}{ \sin \frac{π}{6}}= \dfrac{1}{\frac{1}{2}}=2 \\ \cot \dfrac{π}{6} & = \dfrac{ \cos \frac{π}{6}}{ \sin \frac{π}{6}} \\ &= \dfrac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} =\sqrt{3} \end{align*}\], Find \( \sin t, \cos t, \tan t, \sec t, \csc t,\) and \( \cot t\) when \(t=\frac{π}{3}.\), \(\begin{align} \sin \frac{π}{3} & = \frac{\sqrt{3}}{2} \\ \cos \frac{π}{3} &=\frac{1}{2} \\ \tan \frac{π}{3} &= \sqrt{3} \\ \sec \frac{π}{3} &= 2 \\ \csc \frac{π}{3} &= \frac{2\sqrt{3}}{3} \\ \cot \frac{π}{3} &= \frac{\sqrt{3}}{3} \end{align}\). Now, find the missing side from the given known sides. Download for free at https://openstax.org/details/books/precalculus. Using Even and Odd Trigonometric Functions. The secant of an angle is the same as the secant of its opposite. Because the y-value is equal to the sine of \(t\), and the x-value is equal to the cosine of \(t\), the tangent of angle \(t\) can also be defined as \( \frac{ \sin t}{ \cos t}, \cos t≠0.\) The tangent function is abbreviated as \( \tan.\) The remaining three functions can all be expressed as reciprocals of functions we have already defined. Use properties of even and odd trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships: \[ \cot t= \dfrac{1}{ \tan t}= \dfrac{ \cos t}{ \sin t} \], Example \(\PageIndex{5}\): Using Identities to Evaluate Trigonometric Functions. Use reference angles to find all six trigonometric functions of \(−\frac{7π}{4}\). The results are shown in Table \(\PageIndex{2}\). Jay Abramson (Arizona State University) with contributing authors. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation. Trigonometry The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Solution: 2. sin x = 3/5, x lies in second quadrant. Evaluate the cosecant of \(\frac{5π}{7}\). All along the curve, any two points with opposite x-values have the same function value. 6 - Find the values of the six trigonometric functions... Ch. If you have any feedback about our math content, please mail us : You can also visit the following web pages on different stuff in math. As it turns out, there is an important difference among the functions in this regard. The sign of the sine depends on the y-values in the quadrant where the angle is located. So if the secant of angle t is 2, the secant of \(−t\) is also 2. Figure 4 shows which fu… If cos(t)=1213 cos(t)=1213 and t t is in quadrant IV, as shown in Figure \(\PageIndex{8}\), find the values of the other five trigonometric functions. Evaluate the function at the reference angle. 1. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or \(2π\),will result in the same outputs for these functions. We can find the sine using the Pythagorean Identity, \( \cos ^2 t+ \sin ^2t=1 \), and the remaining functions by relating them to sine and cosine. How do you find the other five trigonometric values given csc b = -6/5 and cot b > 0? Usually, identities can be derived from definitions and relationships we already know. Together they make up the set of six trigonometric functions. We know that. Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting x x equal to the cosine and y y equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. b) tan S=-10/7 and the terminal side of S is in Q2. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. Evaluate trigonometric functions with a calculator. The point \((\frac{\sqrt{2}}{2},−\frac{\sqrt{2}}{2})\) is on the unit circle, as shown in Figure \(\PageIndex{3}\). \[\begin{align*} \sin t &= y=−\dfrac{\sqrt{3}}{2} \\ \cos t &=x =−\dfrac{1}{2} \\ \tan t &= \dfrac{ \sin t}{ \cos t}=\dfrac{−\frac{\sqrt{3}}{2}}{−\frac{1}{2}}= \sqrt{3} \\ \sec t &= \dfrac{1}{\cos t} = \dfrac{1}{−\frac{1}{2}}=−2 \\ \csc t &= \dfrac{1}{\sin t}= \dfrac{1}{−\frac{\sqrt{3}}{2}}=−\dfrac{2\sqrt{3}}{3} \\ \cot t &= \dfrac{1}{ \tan t}=\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3} \end{align*}\]. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. Precalculus: Mathematics for Calcu... 6th Edition. tan (theta) = opp/adj = 2/1 = 2. cot (theta) = adj/opp = 1/2. Use … \[ \begin{align*} \tan(45°) &=\dfrac{ \sin(45°)}{ \cos (45°)} \\ &= \dfrac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} \\ & =1 \end{align*} \], \[\begin{align*} \sec (\dfrac{5π}{6}) &= \dfrac{1}{ \cos (\frac{5π}{6})} \\ &= \dfrac{1}{−\frac{\sqrt{3}}{2}} \\ &= \dfrac{−2\sqrt{3}}{1} \\ &=\dfrac{−2}{\sqrt{3}} \\ &=−\dfrac{2\sqrt{3}}{3} \end{align*}\]. 1 Answer Shwetank Mauria Jul 3, 2016 #sinb=-5/6#, #cosb=-sqrt11/6#, #tanb=5/sqrt11#, #cotb=sqrt11/5# and #secb=-6/sqrt11#. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor \(\frac{π}{180}\) to convert the degrees to radians. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. 400+ SHARES. sin 2 x + cos 2 x = 1. We have already defined the sine and cosine functions of an angle. c) sin S=-5/14 and the terminal side of S is in Q3. Find the value of other five trigonometric function when cos x = -1/2, x lies in third quadrant. The following steps will … The remaining functions can be calculated using identities relating them to sine and cosine. The trigonometric functions are periodic. 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. 5.3: The Other Trigonometric Functions Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent. Measure the angle formed by the terminal side of the given angle and the horizontal axis. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here. In this section, we will investigate the remaining functions. The secant, cotangent, and cosecant are all reciprocals of other functions. We define the trigonometric functions for angles greater than 90° in the following way:By Pythagoras, r=x2+y2\displaystyle{r}=\sqrtx}^{2}+{y}^{2r=x2+y2​. See. The point \((−\frac{\sqrt{3}}{2},\frac{1}{2})\) is on the unit circle, as shown in Figure \(\PageIndex{2}\). View All. Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of  30° (π/6),   45° (π/4), and 60° (π/3). is in quadrant IV, as shown in , find the values of the other five trigonometric functions. The following steps will be useful in the above process. The graph of the function is symmetrical about the y-axis. sin x = 3/5. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis.

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