Chapter 6: Q.52 (page 395)

Find the exact value of each of the remaining trigonometric functions of $θ .$
$cosθ = - \frac{1}{4}, tanθ > 0 .$

Short Answer

Expert verified

The exact values are,

$sinθ = - \frac{\sqrt{15}}{4} cscθ = - \frac{4 \sqrt{15}}{15} secθ = - 4 cotθ = \frac{\sqrt{15}}{15}$

Step by step solution

Step 1. Given Information

Given that $cosθ = - \frac{1}{4}, tanθ > 0$ . To find the exact values for trigonometric functions of angle.

Step 2. Radius of the circle

The angle corresponds to the point $P = (x, y)$ where terminal point is circle. The circle's radius is $r^{2} = x^{2} + y^{2} .$ The values are $cosθ = \frac{x}{r}, x = - 1, r = 4 .$ The tan functions have sine and cosine which occurs in the III and I quadrant. Mainly, $x = - 1 < 0$ must in III quadrant. On solving the value of y,

$y^{2} = r^{2} - x^{2} y^{2} = {(4)}^{2} - {(- 1)}^{2} = 16 - 1 y^{2} = 15 y = \sqrt{15} y = - \sqrt{15}$

Step 3. Finding the exact values

To find the exact values are,

$sinθ = \frac{y}{r} sinθ = - \frac{\sqrt{15}}{4} y^{2} = r^{2} - x^{2} = {(4)}^{2} - {(- 1)}^{2} y^{2} = 15 y = \sqrt{15} in third quadrant, y = - \sqrt{15}$

Step 4. Substitution

On substituting the values to get the exact values,

$cscθ = \frac{r}{y} = - \frac{4}{\sqrt{15}} . \frac{\sqrt{15}}{\sqrt{15}} cscθ = - \frac{4 \sqrt{15}}{15} secθ = \frac{r}{x} = - \frac{4}{1} secθ = - 4 cotθ = \frac{x}{y} = \frac{1}{\sqrt{15}} . \frac{\sqrt{15}}{\sqrt{15}} cotθ = \frac{\sqrt{15}}{15}$

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Find the exact value of each of the remaining trigonometric functions of $θ .$
$cosθ = - \frac{1}{4}, tanθ > 0 .$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Radius of the circle

Step 3. Finding the exact values

Step 4. Substitution

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Find the exact value of each of the remaining trigonometric functions of θ.cosθ=-14,tanθ>0.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Radius of the circle

Step 3. Finding the exact values

Step 4. Substitution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Probability and Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

Find the exact value of each of the remaining trigonometric functions of $θ .$
$cosθ = - \frac{1}{4}, tanθ > 0 .$