Solution: 2. sin x = 3/5, x lies in second quadrant. Click here to see ALL problems on Trigonometry-basics Question 894134 : tan theta=2 find the five other trigonometric function values Found 2 solutions by Theo, Edwin McCravy : For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine. All along the graph, any two points with opposite x-values also have opposite y-values. 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 … Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. \[ \dfrac{ \sec t}{ \tan t}= \csc t \nonumber \], We can use these fundamental identities to derive alternative forms of the Pythagorean Identity, \( \cos ^2 t+ \sin ^2 t=1\). Trig calculator finding sin, cos, tan, cot, sec, csc To find the trigonometric functions of an angle, enter the chosen angle in degrees or radians. Tanθ = Sinθ / Cosθ. The period of the cosine, sine, secant, and cosecant functions is \(2π\). Answer by Edwin McCravy (18456) ( Show Source ): You can put this solution on YOUR website! Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions. We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure \(\PageIndex{7}\). The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. 6 - Find the values of the six trigonometric functions... Ch. Example \(\PageIndex{8}\): Finding the Values of Trigonometric Functions. Even and odd properties can be used to evaluate trigonometric functions. Figure \(\PageIndex{8}\) Solution. 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. Replace the known values in the equation . For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. All the six values are based on a Right Angled Triangle. Solution: It is given that. All along the curve, any two points with opposite x-values have the same function value. 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. The remaining functions can be calculated using identities relating them to sine and cosine. 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 in lllrd Quadrant sin x is … Step 1 : The given given trigonometric ratio has to compared with one of the formulas given below. sin 2 x + cos 2 x = 1. The sine of the positive angle is \(y\). 2. o is the length of the side opposite the angle. Given \( \sin (45°)= \frac{\sqrt{2}}{2}, \cos (45°)= \frac{\sqrt{2}}{2}\), evaluate \( \tan(45°).\). The following steps will be useful in the above process. 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. Figure \(\PageIndex{4}\) shows which functions are positive in which quadrant. 9.2k SHARES. 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. 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. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here. Code to add this calci to your website . A function is said to be even if \(f(−x)=f(x)\) and odd if \(f(−x)=−f(x)\). \[ \begin{align*} \tan \dfrac{π}{6} & = \dfrac{ \sin \frac{π}{6}}{\cos \frac{π}{6}} \\ & = \dfrac{\frac{1}{2} }{\frac{\sqrt{3}}{2}}=\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3} \\ \sec \dfrac{π}{6} &= \dfrac{1}{ \cos \frac{π}{6}} \\ & = \dfrac{1}{\frac{\sqrt{3}}{2}} = \dfrac{2}{\sqrt{3}}= \dfrac{2\sqrt{3}}{3} \\ \csc \dfrac{π}{6} &= \dfrac{1}{ \sin \frac{π}{6}}= \dfrac{1}{\frac{1}{2}}=2 \\ \cot \dfrac{π}{6} & = \dfrac{ \cos \frac{π}{6}}{ \sin \frac{π}{6}} \\ &= \dfrac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} =\sqrt{3} \end{align*}\], Find \( \sin t, \cos t, \tan t, \sec t, \csc t,\) and \( \cot t\) when \(t=\frac{π}{3}.\), \(\begin{align} \sin \frac{π}{3} & = \frac{\sqrt{3}}{2} \\ \cos \frac{π}{3} &=\frac{1}{2} \\ \tan \frac{π}{3} &= \sqrt{3} \\ \sec \frac{π}{3} &= 2 \\ \csc \frac{π}{3} &= \frac{2\sqrt{3}}{3} \\ \cot \frac{π}{3} &= \frac{\sqrt{3}}{3} \end{align}\). We can test each of the six trigonometric functions in this fashion. Formula Used: Sinθ = 1 / Cosecθ Cosθ = 1 / secθ Tanθ = Sinθ / cosθ Cosecθ = 1 / … \[\begin{align*} \sin t &= y=−\dfrac{\sqrt{3}}{2} \\ \cos t &=x =−\dfrac{1}{2} \\ \tan t &= \dfrac{ \sin t}{ \cos t}=\dfrac{−\frac{\sqrt{3}}{2}}{−\frac{1}{2}}= \sqrt{3} \\ \sec t &= \dfrac{1}{\cos t} = \dfrac{1}{−\frac{1}{2}}=−2 \\ \csc t &= \dfrac{1}{\sin t}= \dfrac{1}{−\frac{\sqrt{3}}{2}}=−\dfrac{2\sqrt{3}}{3} \\ \cot t &= \dfrac{1}{ \tan t}=\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3} \end{align*}\]. The tangent of an angle is the ratio of the. \( \cos t=−\frac{8}{17}, \sin t=\frac{15}{17}, \tan t=−\frac{15}{8}\), \( \csc t= \frac{17}{15}, \cot t=−\frac{8}{15} \). That means \(f(−x)=−f(x)\). 9.2k VIEWS. Finding the Value of Trigonometric Functions. Using Even and Odd 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. Access these online resources for additional instruction and practice with other trigonometric functions. Find \( \sin t, \cos t, \tan t, \sec t, \csc t,\) and \( \cot t\). To find the secant of \( 30°\), we could press, \[\mathrm{(for \; a \; scientific \; calculator):\dfrac{1}{30×\frac{π}{180}}COS }\], \[ \mathrm{(for \; a \; graphing \; calculator): \dfrac{1}{cos(\frac{30π}{180})} }\], how to: Given an angle measure in radians, use a scientific calculator to find the cosecant, how to: Given an angle measure in radians, use a graphing utility/calculator to find the cosecant, Example \(\PageIndex{10}\): Evaluating the Cosecant Using Technology. 400+ VIEWS. 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. Trig calculator finding sin, cos, tan, cot, sec, csc. Solution: It is given that. Though sine and cosine are the trigonometric functions most often used, there are four others. Use reference angles to find all six trigonometric functions of \(−\frac{5π}{6}\). If \(\sin (t)= \frac{\sqrt{2}}{2}\) and \(\cos (t)=\frac{\sqrt{2}}{2},\) find \( \sec (t), \csc (t),\tan (t),\) and \( \cot (t)\). Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent Underneath the calculator, six most popular trig functions will appear - three basic ones: sine, cosine and tangent, and their reciprocals: cosecant, secant and cotangent. 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. Figure 4 shows which fu… Legal. If θ is an angle for which the functions are defined, then: The other tutor assumed you meant only the first quadrant. We know that the sine and cosine functions are defined for all real numbers. Solution : tan θ = Opposite side/Adjacent side. 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. In quadrant II, “Smart,” only sine and its reciprocal function, cosecant, are positive. cos 2 … In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent. Use … tan θ = 3. And for tangent and cotangent, only a half a revolution will result in the same outputs. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side. If the graphing utility has degree mode and radian mode, set it to radian mode. Find the value of the other five trigonometric functions : 1:36 400+ LIKES. 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. \\ \text{} & =\dfrac{1}{\sin t} & \text{Simplify and use the identity.} Ex 3.2, 1 Find the values of other five trigonometric functions if cos = – 1/2 , x lies in third quadrant. cot θ = Adjacent side/Opposite side = - √5 /2. Find the values of other five trigonometric functions in Exercises 1 to 5. The graph of the function is symmetrical about the y-axis. Find the value of all the other five trigonometric functions or solve expression If sin(x) cos(x) tan(x) csc(x) sec(x) cot(x) = in Quadrant 1 : 0 <= x < 90 : 0 <= x < pi/2 Quadrant 2 : 90 <= x < 180 : pi/2 <= x < pi Quadrant 3 : 180 <= x < 270 : pi <= x < 3pi/2 Quadrant 4 : 270 <= x < 360 : 3pi/2 <= x < 2pi All possible Quadrants then In Figure \(\PageIndex{1}\), the tangent of angle \(t\) is equal to \(\frac{y}{x},x≠0 \). Use properties of … \( \sin (−\frac{7π}{4})= \frac{\sqrt{2}}{2}, \cos(\frac{−7π}{4})=\frac{\sqrt{2}}{2}, \tan (\frac{−7π}{4})=1,\), \( \sec (\frac{−7π}{4})= \sqrt{2}, \csc (\frac{−7π}{4})= \sqrt{2}, \cot (\frac{−7π}{4})=1 \). Click here to let us know! Don't get them confused! We have explored a number of properties of trigonometric functions. \\ \text{} & = \csc t \end{array}\]. Adopted a LibreTexts for your class? 1. cos x = -1/2, x lies in third quadrant. Ex 3.2, 4 Find the values of other five trigonometric functions if sec x = 13/5 , lies in fourth quadrant. The result is another function that indicates its rate of change (slope) at a particular values of x. Opposite side = 2, Adjacent side = 1 These derivative functions are stated in terms of other trig functions. By using the above formula, find the values of the other trigonometric ratios. The trigonometric functions are periodic. We can write it as. Secant is an even function. Usually, identities can be derived from definitions and relationships we already know. Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of \(\frac{\pi}{3}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{6}\). 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. One form is obtained by dividing both sides by \( \cos ^2 t:\), \[ \begin{align} \dfrac{ \cos ^2 t}{ \cos ^2 t} + \dfrac{ \sin ^2 t}{ \cos ^2 t} & = \dfrac{1}{ \cos ^2 t} \\ 1+ \tan ^2 t & = \sec ^2 t \end{align}\]. The sine function, then, is an odd function. is in quadrant IV, as shown in , find the values of the other five trigonometric functions. 2) Given that sin θ = 5/9 and π/2 < θ < π, find the values of the other 5 trigonometric functions of θ ? For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The secant of an angle is the same as the secant of its opposite. The cosecant, secant, and cotangent are trigonometric functions that are the reciprocals of the sine, cosine, and tangent, respectively. Measure the angle formed by the terminal side of the given angle and the horizontal axis. We also know that for each real number ‘x’, -1 ≤ sinx\sin{x} sinx ≤ 1 and -1 ≤ cosx\cos{x} cosx≤ 1. Finding the Value of Trigonometric Functions. See. The results are shown in Table \(\PageIndex{1}\). Evaluate trigonometric functions with a calculator. If the secant of angle t is 2, what is the secant of \(−t\)? For example, the lengths of months repeat every four years. Cosθ = 1 / Secθ. Example \(\PageIndex{6}\): Using Identities to Simplify Trigonometric Expressions. If the calculator has degree mode and radian mode, set it to radian mode. If trigonometric functions of an angle θ are combined in an equation and the equation is valid for all values of θ, then the equation is known as a trigonometric identity . Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd. tan θ = -2/1. 6 - Find the values of the six trigonometric functions... Ch. 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. If `sin(x)=5/13` then find other trigonometry ; If `cos(x)=12/13` then find other trigonometry ; If `tan(x)=5/12` then find other trigonometry ; If `csc(x)=13/5` then find other trigonometry ; Ex 3.2, 1 Find the values of other five trigonometric functions if cos = – 1/2 , x lies in third quadrant. Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of 30° (π/6), 45° (π/4), and 60° (π/3). See. 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 \(\PageIndex{1}\). The following steps will be useful in the above process. d) sec S=13/11 and the terminal side of angle S is in Q4. 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. b) tan S=-10/7 and the terminal side of S is in Q2. The other form is obtained by dividing both sides by \( \sin ^2 t\): \[ \begin{align} \dfrac{ \cos ^2 t}{ \sin ^2 t}+ \dfrac{ \sin ^2 t}{ \sin ^2 t} &= \dfrac{1}{ \sin ^2 t} \\ \cot ^2 t+1 &= \csc ^2 t \end{align}\], ALTERNATE FORMS OF THE PYTHAGOREAN IDENTITY, Example \(\PageIndex{7}\): Using Identities to Relate Trigonometric Functions. We can write it as. All you have to do now is find the other five trig functions by creating fractions from the three sides. \[ \begin{align} (\dfrac{12}{13})^2+ \sin ^2 t &= 1 \\ \sin ^2 t &=1−(\dfrac{12}{13})^2 \\ \sin ^2 t &=1− \dfrac{144}{169} \\ \sin ^2 t &= \dfrac{25}{169} \\ \sin t &=±\sqrt{\dfrac{25}{169}} \\ \sin t &=±\dfrac{\sqrt{25}}{\sqrt{169}} \\ \sin t &=± \dfrac{5}{13} \end{align} \].
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