Identities can be used to evaluate trigonometric functions. sin 2 x + cos 2 x = 1. The period of the cosine, sine, secant, and cosecant functions is \(2π\). \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), [ "article:topic", "cosecant", "cotangent", "identities", "period", "secant", "tangent", "license:ccby", "showtoc:no", "authorname:openstaxjabramson" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FPrecalculus%2FBook%253A_Precalculus_(OpenStax)%2F05%253A_Trigonometric_Functions%2F5.03%253A_The_Other_Trigonometric_Functions, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), Principal Lecturer (School of Mathematical and Statistical Sciences), 5.2: Unit Circle - Sine and Cosine Functions, Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent, Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent, Using Even and Odd Trigonometric Functions, Recognizing and Using Fundamental Identities, Alternate Forms of the Pythagorean Identity, Evaluating Trigonometric Functions with a Calculator, More Examples of Determining Trig Functions, https://openstax.org/details/books/precalculus, information contact us at info@libretexts.org, status page at https://status.libretexts.org, \(\begin{align} \sin t &=y \\ \sin (−t) &=−y \\ \sin t &≠sin(−t) \end{align}\), \( \begin{align} \cos t &=x \\ \cos (−t)=x \\ \cos t &= \cos (−t) \end{align}\), \(\begin{align} \tan (t) &= \frac{y}{x} \\ \tan (−t) &=−\frac{y}{x} \\ \tan t &≠ \tan (−t) \end{align}\), \(\begin{align} \sec t &= \frac{1}{x} \\ \sec (−t) &= \frac{1}{x} \\ \sec t &= \sec (−t) \end{align}\), \( \begin{align} \csc t &= \frac{1}{y} \\ \csc (−t) &= \frac{1}{−y} \\ \csc t &≠ \csc (−t) \end{align}\), \( \begin{align} \cot t &= \frac{x}{y} \\ \cot (−t) &= \frac{x}{−y} \\ \cot t & ≠ \cot (−t) \end{align}\), \( \cot t= \frac{1}{\tan t}= \frac{\cos t}{ \sin t}\). This matches the result of calculation: \((4)^2=(−4)^2,(−5)^2=(5)^2\), and so on. The other tutor assumed you meant only the first quadrant. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. Serenity E. asked • 01/07/14 Find the value of ALL the other five trigonometric functions, given tanx= 4/11, secx<0. All along the curve, any two points with opposite x-values have the same function value. Stewart + 5 others. The period of the tangent and cotangent functions is \(π\). Step 1 : The given given trigonometric ratio has to compared with one of the formulas given below. Now consider the function \(f(x)=x^3\), shown in Figure \(\PageIndex{6}\). 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 : To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. The only difference is that now x or y (or both) can be negative because our angle can now be in any quadrant. 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 \( \sec t= \sqrt{2},\csc t=\sqrt{2}, \tan t=1, \cot t=1\). 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. We have explored a number of properties of trigonometric functions. In second quadrant, sinx and cosecx are positive and tanx, cotx and cosx & secx are negative. Each of the functions can be differentiated in calculus. 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. 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. Figure \(\PageIndex{8}\) Solution. The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine. All you have to do now is find the other five trig functions by creating fractions from the three sides. 1. Measure the angle formed by the terminal side of the given angle and the horizontal axis. Finding the Value of Trigonometric Functions. Usually, identities can be derived from definitions and relationships we already know. In quadrant II, “Smart,” only sine and its reciprocal function, cosecant, are positive. The value of one of the trigonometric functions of an angle θ is given along with the information about the quadrant in which θ lies, Find the values of the other five trigonometric functions of θ: cos2 θ + sin2 θ = 1. Since \(−\frac{5π}{6}\) is in the third quadrant, where both \(x\) and \(y\) are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive. tan θ = -2/1. If If x and x are in the second quadrant, find the other five trigonometric functions. So \(f(x)=x^3\) is an odd function, one such that two inputs that are opposites have outputs that are also opposites. 400+ SHARES. Trigonometry functions calculator that finds the values of Sin, Cos and Tan based on the known values. The following steps will be useful in the above process. First we use the Pythagorean trigonometric identity: sin^2 x + cos^2 x =1 sin^2 x +(3/5)^2=1 sin^2 x +9/25=1 sin^2 x =1-9/25 sin^2 x=16/25 sin x=+-4/5 Now, if we knew more … If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation. Question 2 : Find the values of other five trigonometric functions for the following: tan θ = -2, θ lies in the II quadrant. Together they make up the set of six trigonometric functions. 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. Finding Trigonometric Values Given One Trigonometric Value/Other Info Example 2. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. The trigonometric functions are periodic. We can derive some useful identities from the six trigonometric functions. This is the reference angle. 6 - Find the values of the six trigonometric functions... Ch.
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