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Expert-verified Found in: Page 661 ### Algebra 1

Book edition Student Edition
Author(s) Carter, Cuevas, Day, Holiday, Luchin
Pages 801 pages
ISBN 9780078884801 # Find the values of the three trigonometric ratios of $\mathbit{A}$. The three trigonometric ratios of $A$ are as follows.

role="math" localid="1647948666906" $\begin{array}{c}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{A}\mathbf{=}\frac{\mathbf{4}}{\mathbf{5}}\\ \\ \mathbf{c}\mathbf{o}\mathbf{s}\mathbf{A}\mathbf{=}\frac{\mathbf{3}}{\mathbf{5}}\\ \\ \mathbf{t}\mathbf{a}\mathbf{n}\mathbf{A}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\end{array}$

See the step by step solution

## Step 1. Define the three trigonometric ratios.

If $\theta$ is any angle in a given right angle triangle, then the three trignometric ratios are,

$\begin{array}{c}sin\theta =\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}opposite\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle \theta \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}hypotenuse}}\\ \\ cos\theta =\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}adjacent\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle \theta \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}hypotenuse}}\\ \\ tan\theta =\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}opposite\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle \theta }{\text{ength\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}adjacent\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle \theta \text{\hspace{0.17em}\hspace{0.17em}}}\end{array}$

## Step 2. Calculate the value of three trigonometric ratios of angle A.

Observe the figure. From the figure, $AB$ is hypotenuse (side opposite to 90 degree) and length of $AB$ is 5 units.

$CB$ is the side opposite to angle $A$ and length of $CB$ is 4 units.

$AC$ is the side adjacent to angle $A$ and length of $AC$ is 3 units.

$\begin{array}{c}sinA=\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}opposite\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle A\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}hypotenuse}}\\ =\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}CB}}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}hypotenuse}}\\ =\frac{4}{5}\\ \\ cosA=\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}adjacent\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle A\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}hypotenuse}}\\ =\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}AC}}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}hypotenuse}}\\ =\frac{3}{5}\\ \\ tanA=\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}opposite\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle A}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}adjacent\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}\angle A\text{\hspace{0.17em}\hspace{0.17em}}}\\ =\frac{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}CB}}{\text{length\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}AC}}\\ =\frac{4}{3}\end{array}$

## Step 3. State the conclusion.

The three trigonometric ratios are $\mathrm{sin}A$,$\mathrm{cos}A$, $\mathrm{tan}A$and their values are as follows.

$\begin{array}{c}\mathrm{sin}A=\frac{4}{5}\\ \mathrm{cos}A=\frac{3}{5}\\ \mathrm{tan}A=\frac{4}{3}\end{array}$ ### Want to see more solutions like these? 