evaluating a composition of functions in exercises 51 - 62, find the exact value of the expression, if…
evaluating a composition of functions in exercises 51 - 62, find the exact value of the expression, if possible. 52. cos(arcsin 4/5) 54. sin(cos^(-1) √5) 56. cscarctan(-5/12) 58. secarccos(-1/4) 60. cot(arctan 5/8) 62. tansin^(-1)(-√2/2) writing an expression in exercises 63 - 72, write an algebraic expression that is equivalent to the given expression. 64. sin(arctan x) 66. sec(arctan 3x) 68. cscarccos(x - 1) 70. cot(arctan 1/x)
Answer
Explanation:
Step1: Let $\theta=\arcsin\frac{4}{5}$
By definition, $\sin\theta = \frac{4}{5}$ and $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$.
Step2: Use the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$
We get $\cos^{2}\theta=1 - \sin^{2}\theta$. Substituting $\sin\theta=\frac{4}{5}$, we have $\cos^{2}\theta=1 - (\frac{4}{5})^{2}=1-\frac{16}{25}=\frac{9}{25}$.
Step3: Determine the sign of $\cos\theta$
Since $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$ and $\sin\theta=\frac{4}{5}>0$, $\theta$ is in the first - quadrant where $\cos\theta>0$. So $\cos\theta=\frac{3}{5}$. Thus, $\cos(\arcsin\frac{4}{5})=\frac{3}{5}$.
Answer:
$\frac{3}{5}$