QuestionSketch the least positive angle $\theta$ for the line $-7 x + 8 y = 0$ (with $x \geq 0$ ) and find its six trig functions.

Studdy Solution

STEP 1

Assumptions1. The equation of the line is $-7x +8y =0$ . The restriction on $x$ is $x \geq0$
3. The angle $\theta$ is in standard position, which means its vertex is at the origin and its initial side is along the positive x-axis4. We need to find the least positive angle $\theta$ and the six trigonometric functions of $\theta$

STEP 2

First, we need to rewrite the equation of the line in slope-intercept form, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
$-7x +8y =0$

STEP 3

Rearrange the equation to solve for $y$ .
$y = \frac{7}{8}x$

STEP 4

Now, we can see that the line passes through the origin (0,0) and has a positive slope of $\frac{7}{8}$ . This means the line makes an angle with the positive x-axis.

STEP 5

The slope of the line is the tangent of the angle it makes with the positive x-axis. Therefore, we can find the angle $\theta$ by taking the arctangent of the slope.
$\theta = \arctan\left(\frac{7}{8}\right)$

STEP 6

Now, we need to find the six trigonometric functions of $\theta$ . These are sine, cosine, tangent, cosecant, secant, and cotangent.

STEP 7

The line equation $y = \frac{7}{}x$ gives us a right triangle with opposite side7 and adjacent side when $x$ is positive. The hypotenuse of this triangle can be found using the Pythagorean theorem.
$Hypotenuse = \sqrt{7^2 +^2}$

STEP 8

Calculate the hypotenuse.
$Hypotenuse = \sqrt{7^2 +8^2} = \sqrt{113}$

STEP 9

Now we can find the six trigonometric functions.
The sine of $\theta$ is the ratio of the opposite side to the hypotenuse.
$\sin(\theta) = \frac{7}{\sqrt{113}}$

STEP 10

The cosine of $\theta$ is the ratio of the adjacent side to the hypotenuse.
$\cos(\theta) = \frac{8}{\sqrt{113}}$

STEP 11

The tangent of $\theta$ is the ratio of the opposite side to the adjacent side.
$\tan(\theta) = \frac{7}{8}$

STEP 12

The cosecant of $\theta$ is the reciprocal of the sine of $\theta$ .
$\csc(\theta) = \frac{\sqrt{113}}{7}$

STEP 13

The secant of $\theta$ is the reciprocal of the cosine of $\theta$ .
$\sec(\theta) = \frac{\sqrt{113}}{8}$

STEP 14

The cotangent of $\theta$ is the reciprocal of the tangent of $\theta$ .
$\cot(\theta) = \frac{8}{7}$ So, the least positive angle $\theta$ is $\arctan\left(\frac{7}{8}\right)$ , and the six trigonometric functions of $\theta$ are $\sin(\theta) = \frac{7}{\sqrt{113}}$ , $\cos(\theta) = \frac{8}{\sqrt{113}}$ , $\tan(\theta) = \frac{7}{8}$ , $\csc(\theta) = \frac{\sqrt{113}}{7}$ , $\sec(\theta) = \frac{\sqrt{113}}{8}$ , and $\cot(\theta) = \frac{8}{7}$ .

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