Question

Find the values of the three trigonometric ratios of $\mathit{A}$.

Short answer

The three trigonometric ratios of $A$are as follows.

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

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,

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

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.

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

Step 3. State the conclusion.

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

$\begin{array}{c}\sin A=\frac{4}{5}\\ \cos A=\frac{3}{5}\\ \tan A=\frac{4}{3}\end{array}$