Chapter 69, Problem 13A

To determine

To sketch:

A right triangle for the given angle and the sides of the triangle are to be labelled as positive or negative.

Label the sides of the triangle as positive or negative.

To find: The reference angles and functions of the angles.

The value of sine, cosine, tangent, cotangent, cosecant, secant for a given angle, up to 4 decimals.

Answer to Problem 13A

Reference angle is 28°50'.

Sin 208°50' is -0.4822. It is a negative function.

Cos 208°50' is -0.876. It is a negative function.

Tan 208°50' is 0.5504. It is a positive function.

Cot 208°50' is 1.8167. It is a positive function.

Sec 208°50' is -1.1414. It is a negative function.

Cosec 208°50' is -2.073. It is a negative function.

Explanation of Solution

Given Information:

θ= 208°50'

In Quadrant 3, y is negative and x is negative.

In the above triangle, r is the hypotenuse, the adjacent side x is negative, the opposite side y is negative.

  $\begin{array}{l}Referenceangle\angle \alpha =208°50’-180°\hfill \\ \angle \alpha =28.83°\hfill \end{array}$

  $\begin{array}{l}\sin \alpha =\frac{-y}{r}\\ Itisanegativefunction.\\ \sin 208°50’=\sin (208°50’-180°)\\ \sin 208°50’=-\sin 28.83°\\ \sin 208°50’=-0.4822\\ \cos \alpha =\frac{-x}{r}\\ Itisanegativefunction.\\ \cos 208°50’=\cos (208°50’-180°)\\ \cos 208°50’=-\cos 28.83°\\ \cos 208°50’=-0.876\\ \tan \alpha =\frac{-y}{-x}\\ Itisapositivefunction\\ \tan 208°50’=\tan (208°50’-180°)\\ \tan 208°50’=\tan 28.83°\\ \tan 208°50’=0.5504\\ \cot \alpha =\frac{-x}{-y}\\ Itisapositivefunction\\ \cot 208°50’=\cot (208°50’-180°)\\ \cot 208°50’=-\cot 28.83°\\ \cot 208°50’=1.8167\\ \sec \alpha =\frac{r}{-x}\\ Itisanegativefunction.\\ \sec 208°50’=\sec (208°50’-180°)\\ \sec 208°50’=\sec 28.83°\\ \sec 208°50’=-1.1414\\ \cos ec\alpha =\frac{r}{-y}\\ Itisanegativefunction.\\ \mathrm{co}\sec 208°50’=\mathrm{co}\sec (208°50’-180°)\\ \mathrm{co}\sec 208°50’=-\mathrm{co}\sec 28.83°\\ \mathrm{co}\sec 208°50’=-2.0737\end{array}$

Conclusion:

Reference angle is 28°50'.

Sin 208°50' is -0.4822. It is a negative function.

Cos 208°50' is -0.876. It is a negative function.

Tan 208°50' is 0.5504. It is a positive function.

Cot 208°50' is 1.8167. It is a positive function.

Sec 208°50' is -1.1414. It is a negative function.

Cosec 208°50' is -2.073. It is a negative function.