ANSWERS
• sin(θ) = -4/5
,
• cos(θ) = -3/5
,
• tan(θ) = 4/3
,
• sec(θ) = -5/3
,
• csc(θ) = -5/4
,
• cot(θ) = 3/4
EXPLANATION
We have to find the six trigonometric functions of θ: sin(θ), cos(θ), tan(θ), sec(θ), csc(θ), and cot(θ).
To do so, we have to use the right triangle formed by the given vector and its components,
Using this triangle, we can evaluate the trigonometric functions of angle α, which are the same for angle θ. The hypotenuse of the triangle is,
[tex]r=\sqrt{(-6)^2+(-8)^2}=\sqrt{36+64}=\sqrt{100}=10[/tex]
The first three trigonometric functions are,
[tex]\begin{gathered} \sin\theta=\frac{opposite}{hypotenuse} \\ \\ \cos\theta=\frac{adjacent}{hypotenuse} \\ \\ \tan\theta=\frac{opposite}{adjacent} \end{gathered}[/tex]
The opposite side is -8, the adjacent side is -6, and the hypotenuse is 10,
[tex]\begin{gathered} \sin\theta=\frac{-8}{10}=-\frac{4}{5} \\ \\ \cos\theta=\frac{-6}{10}=-\frac{3}{5} \\ \\ \tan\theta=\frac{-8}{-6}=\frac{4}{3} \end{gathered}[/tex]
Now, the secant is the reciprocal of the cosine, the cosecant is the reciprocal of the sine, and the cotangent is the reciprocal of the tangent,
[tex]\begin{gathered} \sec\theta=\frac{1}{\cos\theta} \\ \\ \csc\theta=\frac{1}{\sin\theta} \\ \\ \cot\theta=\frac{1}{\tan\theta} \end{gathered}[/tex]
These are known values, so we have,
[tex]\begin{gathered} \sec\theta=\frac{1}{-\frac{3}{5}}=-\frac{5}{3} \\ \\ \csc\theta=\frac{1}{-\frac{4}{5}}=-\frac{5}{4} \\ \\ \cot\theta=\frac{1}{\frac{4}{3}}=\frac{3}{4} \end{gathered}[/tex]
Hence, the values of the six trigonometric functions are:
• sin(θ) = -4/5
,
• cos(θ) = -3/5
,
• tan(θ) = 4/3
,
• sec(θ) = -5/3
,
• csc(θ) = -5/4
,
• cot(θ) = 3/4
