Questions: 5. (5 pts.) Find the exact value for each of the indicated trigonometric functions of θ. Show your work. cos (θ)= cos (θ)= tan (θ)= sec (θ)= cot (θ)=

Solution Steps

Using the Pythagorean theorem ($a^2 + b^2 = c^2$), where $a$ and $b$ are the legs of the right triangle and $c$ is the hypotenuse, we can find the length of the side opposite to $\theta$. Let's call this side $x$. We are given that one leg is 4 and the hypotenuse is 7, so we have:

$4^2 + x^2 = 7^2$

$16 + x^2 = 49$

$x^2 = 33$

$x = \sqrt{33}$

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{7}$

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

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

Final Answer: