Question

Use the table of trigonometric ratios to answer the following questions

Answer 1

Step-by-step explanation

Step 1: We draw the triangle described in the word problem. Since the triangle can be any similar right triangle, we can choose the measure of the sides under the condition that the triangle will be a right triangle and one of its angles measures 28°. For example, the measure of one side can be 5 units.

Step 2: We find the measure of the opposite leg and the hypotenuse.

• Opposite leg: Since it is a right triangle, we can use the trigonometric ratio tan(θ).

`\tan(\theta)=\frac{\text{ Opposite leg}}{\text{ Adjacent leg}}`

Then, we have:

`\begin{gathered} \tan(28°)=\frac{\text{ Opposite leg}}{5} \\ \text{ Multiply by 5 from both sides} \\ 5\cdot\tan(28\degree)=\frac{\text{Opposte leg}}{5}\cdot5 \\ 2.66\approx\text{ Opposite leg} \\ \text{ The symbol }\approx\text{ is read 'approximately'.} \end{gathered}`

• Hypotenuse: Since it is a right triangle, we can use the trigonometric ratio cos(θ).

`\cos(\theta)=\frac{\text{ Adjacent leg}}{\text{ Hypotenuse}}`

Then, we have:

`\begin{gathered} \cos(28°)=\frac{5}{\text{ Hypotenuse}} \\ \cos(28°)\cdot\text{ Hypotenuse }=\frac{5}{\text{ Hypotenuse}}\cdot\text{ Hypotenuse} \\ \cos(28°)\cdot\text{ Hypotenuse }=5 \\ \frac{\cos(28\degree)\cdot\text{ Hypotenuse}}{\cos(28\degree)}=(5)/(\cos(28\degree)) \\ \text{ Hypotenuse }\approx5.66 \end{gathered}`

Step 3: We find the trigonometric ratios sine, cosine and tangent.

• Sine

`\begin{gathered} \sin(\theta)=\frac{\text{ Opposite leg}}{\text{ Hypotenuse}} \\ \sin(28\degree)\approx(2.66)/(5.66) \\ \sin(28\degree)\approx0.47 \end{gathered}`

• Cosine

`\begin{gathered} \cos(\theta)=\frac{\text{ Adjacent leg}}{\text{ Hypotenuse}} \\ \cos(28\degree)\approx(5)/(5.66) \\ \cos(28\degree)\approx0.88 \end{gathered}`

• Tangent

`\begin{gathered} \tan(\theta)=\frac{\text{ Opposite leg}}{\text{ Adjacent leg}} \\ \tan(28\degree)\approx(2.66)/(5) \\ \tan(28\degree)\approx0.53 \end{gathered}`Answer
`\begin{gathered} \sin(28\degree)\approx0.47 \\ \cos(28\degree)\approx0.88 \\ \tg(28\degree)\approx0.53 \end{gathered}`