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Answer 1

The value of the expression $2\cos(t) + \tan^2(t)$ is 4 and the exact value of $\sin^{-1}(\sin(\frac{5\pi}{3}))$ is $\frac{5\pi}{3}$

How to determine the trigonometry expression?

The point on the unit circle is given as:

$P = (\frac 12, \frac{\sqrt{3}}2)$

A point on a unit circle is represented as: (x,y), such that:

cos(t) = x and sin(t) = y.

This means that:

$\cos(t) = \frac 12$

$\sin(t) = \frac{\sqrt 3}{2}$

Calculate tan(t) using:

$\tan(t) = \frac{\sin(t)}{\cos(t)}$

So, we have:

$\tan(t) = \frac{\frac{\sqrt 3}{2}}{1/2}$

Evaluate

$\tan(t) = \sqrt 3$

The expression is then calculated as:

$2\cos(t) + \tan^2(t) = 2 * \frac12 + (\sqrt 3)^2$

Evaluate each term

$2\cos(t) + \tan^2(t) = 1 + 3$

Evaluate the sum

$2\cos(t) + \tan^2(t) = 4$

Hence, the value of the expression $2\cos(t) + \tan^2(t)$ is 4

How to solve the arcsin expression?

The expression is given as:

$\sin^{-1}(\sin(\frac{5\pi}{3}))$

As a general rule, the arc sine of sine x is x.

This means that:

$\sin^{-1}(\sin(\frac{5\pi}{3})) = \frac{5\pi}{3}$

Hence, the exact value of $\sin^{-1}(\sin(\frac{5\pi}{3}))$ is $\frac{5\pi}{3}$

Read more about trigonometry expressions at:

brainly.com/question/8120556

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