Question

find the sine, cosine, and tangent of ∠q. simplify your answers and write them as proper numbers. sin(q) = cos(q) =

Explanation:

Step1: Find the hypotenuse using Pythagorean theorem

Let the hypotenuse be $PQ$. By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = 39$, $b = 80$. So $PQ=\sqrt{39^{2}+80^{2}}=\sqrt{1521 + 6400}=\sqrt{7921}=89$.

Step2: Calculate $\sin(Q)$

The definition of sine in a right - triangle is $\sin(Q)=\frac{\text{opposite}}{\text{hypotenuse}}$. The side opposite to $\angle Q$ is $PR = 80$ and the hypotenuse $PQ = 89$. So $\sin(Q)=\frac{80}{89}$.

Step3: Calculate $\cos(Q)$

The definition of cosine in a right - triangle is $\cos(Q)=\frac{\text{adjacent}}{\text{hypotenuse}}$. The side adjacent to $\angle Q$ is $RQ = 39$ and the hypotenuse $PQ = 89$. So $\cos(Q)=\frac{39}{89}$.

Step4: Calculate $\tan(Q)$

The definition of tangent in a right - triangle is $\tan(Q)=\frac{\text{opposite}}{\text{adjacent}}$. The side opposite to $\angle Q$ is $PR = 80$ and the side adjacent to $\angle Q$ is $RQ = 39$. So $\tan(Q)=\frac{80}{39}$.

Answer:

$\sin(Q)=\frac{80}{89}$, $\cos(Q)=\frac{39}{89}$, $\tan(Q)=\frac{80}{39}$