Question

I need help with this.. having trouble solving it’s a practice from my ACT prep guide

Answer 1

Given the question in the image, the following are the solution steps to answer the question

STEP 1: Draw the described triangle

STEP 2: Find the third side

`\begin{gathered} \text{third side is the hypotenuse} \\ \text{opposite}=8,\text{adjacent}=4,\text{hypotenuse}=\text{?} \\ By\text{ pythagorean theorem,} \\ \text{hypotenuse}^2=opposite^2+adjacent^2 \\ \text{hypotenuse}^2=8^2+4^2 \\ \text{hypotenuse}^2=64+16=80 \\ \text{hypotenuse}=\sqrt[]{80}=\sqrt[]{16*5}=\sqrt[]{16}*\sqrt[]{5}=4*\sqrt[]{5}=4\sqrt[]{5} \end{gathered}`

STEP 3: Redraw the complete triangles

STEP 4: Find the trigonometric ratio for csc B

The cosecant of angle B in a right triangle is equal to the length of the hypotenuse divided by the opposite side. To solve csc, simply enter the length of the hypotenuse and opposite side, then solve.

`\begin{gathered} \csc B\text{ stands for Cosec B} \\ \csc B=(1)/(\sin\theta)=(hypotenuse)/(opposite) \\ \text{For the angle B,} \\ \text{hypoteunse}=4\sqrt[]{5},opposite=8,adjacent=4 \\ By\text{ substituting into the formula, we have} \\ \csc B=\frac{4\sqrt[]{5}}{8}=\frac{1\sqrt[]{5}}{2}=\frac{\sqrt[]{5}}{2} \end{gathered}`

STEP 5: Find the trigonometric ratio for cot B

The cotangent formula for an angle θ is: cot θ = (Adjacent side) / (Opposite side)

`\begin{gathered} \text{cot B stands for cotangent B} \\ \cot B=\frac{\text{adjacent}}{\text{opposite}} \\ \text{opposite}=8,\text{adjacent}=4 \\ By\text{ substitution.} \\ \cot B=(4)/(8)=(1)/(2) \end{gathered}`

STEP 5: Find the trigonometric ratio for cos A

`\begin{gathered} \cos A\text{ stands for Cosine A} \\ \cos A=\frac{\text{adjacent}}{\text{hypotenuse}} \\ \text{adjacent}=8,\text{hypotenuse}=4\sqrt[]{5} \\ By\text{ substitution,} \\ \cos A=\frac{8}{4\sqrt[]{5}}=\frac{2}{\sqrt[]{5}} \\ By\text{ rationalization of surds, we have;} \\ \frac{2}{\sqrt[]{5}}*\frac{\sqrt[]{5}}{\sqrt[]{5}}=\frac{2*\sqrt[]{5}}{\sqrt[]{5}*\sqrt[]{5}}=\frac{2\sqrt[]{5}}{\sqrt[]{25}} \\ \cos A=\frac{2\sqrt[]{5}}{5} \end{gathered}`