Question

quiz 2b: trig angles & ratios and special right triangles for #1 - 3: all answers should be given in either simplified fraction or radical form if not an integer (no decimals!) 1. given: cosθ = 4/7 (7 points) a) draw and label a right triangle using the given information, including θ and the right angle. b) find the missing side length. c) find the remaining five trig ratios for θ. sinθ = tanθ = cscθ = secθ = cotθ =

Explanation:

Step1: Recall cosine definition

We know that $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Given $\cos\theta = \frac{4}{7}$, we can draw a right - triangle where the adjacent side to the angle $\theta$ has length $a = 4$ and the hypotenuse has length $c=7$.

Step2: Use Pythagorean theorem to find the missing side

By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = 4$ and $c = 7$. Then $b=\sqrt{c^{2}-a^{2}}=\sqrt{7^{2}-4^{2}}=\sqrt{49 - 16}=\sqrt{33}$.

Step3: Find $\sin\theta$

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{\sqrt{33}}{7}$.

Step4: Find $\tan\theta$

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{\sqrt{33}}{4}$.

Step5: Find $\csc\theta$

$\csc\theta=\frac{1}{\sin\theta}=\frac{7}{\sqrt{33}}=\frac{7\sqrt{33}}{33}$.

Step6: Find $\sec\theta$

$\sec\theta=\frac{1}{\cos\theta}=\frac{7}{4}$.

Step7: Find $\cot\theta$

$\cot\theta=\frac{1}{\tan\theta}=\frac{4}{\sqrt{33}}=\frac{4\sqrt{33}}{33}$.

Answer:

a) Draw a right - triangle with an angle $\theta$, the adjacent side to $\theta$ of length 4, the hypotenuse of length 7, and the right - angle.
b) $\sqrt{33}$
c) $\sin\theta=\frac{\sqrt{33}}{7}$, $\tan\theta=\frac{\sqrt{33}}{4}$, $\csc\theta=\frac{7\sqrt{33}}{33}$, $\sec\theta=\frac{7}{4}$, $\cot\theta=\frac{4\sqrt{33}}{33}$