Question

Find the values of the trigonometric functions of ∅ from the information given.tan(∅) = -2, sin(∅) > 0

Answer 1

Solution

$\tan \emptyset=-2,\sin \emptyset>0$

Call tan ∅ = t = -2, Use the trig identity:

$\cos ^2\emptyset=\frac{1}{1+t^2}$

Since tan ∅ = -3 is negative and sin ∅ > 0, then ∅ should be in quadrant II

$\begin{gathered} \cos ^2\emptyset=\frac{1}{1+4}=\frac{1}{5}=0.2 \\ \cos ^2\emptyset=0.2 \\ \cos \emptyset=-0.45 \end{gathered}$$\begin{gathered} \sin ^2\emptyset=1-\cos ^2\emptyset=1-0.2=0.8 \\ \sin ^2\emptyset=0.8 \\ \sin \emptyset=0.89,\text{ }\mleft(QuadrantII\mright) \end{gathered}$$\cot \emptyset=-\frac{1}{2}$$co\sec \emptyset=\frac{1}{\sin\emptyset}=\frac{1}{0.894}=1.12$$\sec \emptyset=\frac{1}{\cos\emptyset}=\frac{1}{-0.447}=-2.24$