Chapter 6.3, Problem 1E

If the angle θ is in standard position and P(x, y) is a point on the terminal side of θ, and r is the distance from the origin to P, then

$\sin \theta =\frac{\boxed{}}{\boxed{}}\cos \theta =\frac{\boxed{}}{\boxed{}}\tan \theta =\frac{\boxed{}}{\boxed{}}$

To determine

To evaluate: The value of trigonometric function $\sin \theta$, $\cos \theta$ and $\tan \theta$.

Answer to Problem 1E

The value of trigonometric function $\sin \theta$ is $\frac{b}{r}$, $\cos \theta$ is $\frac{a}{r}$ and $\tan \theta$ is $\frac{b}{a}$.

Explanation of Solution

Given:

The angle $\theta$ in the standard position and $P\left(x,y\right)$ is a point in the terminal side of $\theta$ and r is the distance from the origin to the point P.

Calculation:

The Figure which shows the terminal side and the angle $\theta$ is,

Figure (1)

From the Figure (1), the value of $\sin \theta$ is,

$\sin \theta =\frac{\text{perpendicular}}{\text{height}}$ (1)

Substitute b for perpendicular and r for height in equation (1),

$\sin \theta =\frac{b}{r}$

From the Figure (1), the value of $\cos \theta$ is,

$\cos \theta =\frac{\text{base}}{\text{height}}$ (2)

Substitute a for base and r for height in equation (2),

$\cos \theta =\frac{a}{r}$

From the figure (1), the value of $\tan \theta$ is,

$\tan \theta =\frac{\text{perpendicular}}{\text{base}}$ (3)

Substitute b for perpendicular and a for base in equation (3),

$\tan \theta =\frac{b}{a}$

The value of trigonometric function $\sin \theta$ is $\frac{b}{r}$, $\cos \theta$ is $\frac{a}{r}$ and $\tan \theta$ is $\frac{b}{a}$.