7.4 The other trigonometric functions (Page 2/14)

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The point $(\frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2})$ is on the unit circle, as shown in [link] . Find $\sin t, \cos t, \tan t, \sec t, \csc t,$ and $\cot t .$

This is an image of a graph of circle with angle of t inscribed with radius 1. Point of (square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

$\sin t = - \frac{\sqrt{2}}{2}, \cos t = \frac{\sqrt{2}}{2}, \tan t = - 1, s e c t = \sqrt{2}, \csc t = - \sqrt{2}, \cot t = - 1$

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Finding the trigonometric functions of an angle

Find $\sin t, \cos t, \tan t, \sec t, \csc t,$ and $\cot t .$ when $t = \frac{π}{6} .$

We have previously used the properties of equilateral triangles to demonstrate that $\sin \frac{π}{6} = \frac{1}{2}$ and $\cos \frac{π}{6} = \frac{\sqrt{3}}{2} .$ We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.

\begin{matrix} tan \frac{π}{6} & = \frac{sin \frac{π}{6}}{cos \frac{π}{6}} \\ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} & = \frac{1}{\sqrt{3}} & = \frac{\sqrt{3}}{3} \\ sec \frac{π}{6} & = \frac{1}{cos \frac{π}{6}} \\ = \frac{1}{\frac{\sqrt{3}}{2}} & = \frac{2}{\sqrt{3}} & = \frac{2 \sqrt{3}}{3} \\ \csc \frac{π}{6} & = \frac{1}{\sin \frac{π}{6}} & = \frac{1}{\frac{1}{2}} & = 2 \\ cot \frac{π}{6} & = \frac{cos \frac{π}{6}}{sin \frac{π}{6}} \\ = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} & = \sqrt{3} \end{matrix}

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Try It

Find $\sin t, \cos t, \tan t, \sec t, \csc t,$ and $\cot t .$ when $t = \frac{π}{3} .$

$\begin{matrix} \sin \frac{π}{3} = \frac{\sqrt{3}}{2} \\ \cos \frac{π}{3} = \frac{1}{2} \\ \tan \frac{π}{3} = \sqrt{3} \\ \sec \frac{π}{3} = 2 \\ \csc \frac{π}{3} = \frac{2 \sqrt{3}}{3} \\ \cot \frac{π}{3} = \frac{\sqrt{3}}{3} \end{matrix}$

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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 $x$ equal to the cosine and $y$ equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in [link] .

Angle	$0$	$\frac{π}{6}, or 30°$	$\frac{π}{4}, or 45°$	$\frac{π}{3}, or 60°$	$\frac{π}{2}, or 90°$
Cosine	1	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	0
Sine	0	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	1
Tangent	0	$\frac{\sqrt{3}}{3}$	1	$\sqrt{3}$	Undefined
Secant	1	$\frac{2 \sqrt{3}}{3}$	$\sqrt{2}$	2	Undefined
Cosecant	Undefined	2	$\sqrt{2}$	$\frac{2 \sqrt{3}}{3}$	1
Cotangent	Undefined	$\sqrt{3}$	1	$\frac{\sqrt{3}}{3}$	0

Using reference angles to evaluate tangent, secant, cosecant, and cotangent

We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. 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. [link] shows which functions are positive in which quadrant.

To help remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “ A ,” a ll of the six trigonometric functions are positive. In quadrant II, “ S mart,” only s ine and its reciprocal function, cosecant, are positive. In quadrant III, “ T rig,” only t angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “ C lass,” only c osine and its reciprocal function, secant, are positive.

This image is a graph of circle with each quadrant labeled. Under quadrant I, labels for sin t, cos t, tan t, sec t, csc t, and cot t. Under quadrant II, labels for sin t and csc t. Under quadrant III, labels for tan t and cot t. Under quadrant IV, labels for cos t, sec t. — The trigonometric functions are each listed in the quadrants in which they are positive.

How To

Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.

Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle.
Evaluate the function at the reference angle.
Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.

Using reference angles to find trigonometric functions

Use reference angles to find all six trigonometric functions of $- \frac{5 π}{6} .$

The angle between this angle’s terminal side and the x -axis is $\frac{π}{6},$ so that is the reference angle. Since $- \frac{5 π}{6}$ 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.

$\begin{matrix} cos (- \frac{5 π}{6}) & = - \frac{\sqrt{3}}{2}, sin (- \frac{5 π}{6}) & = - \frac{1}{2}, tan (\frac{5 π}{6}) & = \frac{\sqrt{3}}{3}, \\ sec (- \frac{5 π}{6}) & = - \frac{2 \sqrt{3}}{3}, csc (- \frac{5 π}{6}) & = −2, cot (- \frac{5 π}{6}) & = \sqrt{3} \end{matrix}$

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Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

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Desalegn

what is ecnomics

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this is the study of how the society manages it's scarce resources

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macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

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etc

husaini

difference between firm and industry

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what's the difference between a firm and an industry

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firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

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factors influencing supply

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scares means__________________ends resources. unlimited

Jan

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Jan

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out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

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economic overview notes

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identify a demand and a supply curve

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i don't know

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there's a difference

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Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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