Chapter 6: Q. 51 (page 395)

Find the exact value of each of the remaining trigonometric functions of $θ .$
role="math" localid="1646521470980" $sinθ = \frac{2}{3}, tanθ < 0$

Short Answer

Expert verified

The exact values are,

$cosθ = - \frac{\sqrt{5}}{3} tanθ = - \frac{2 \sqrt{5}}{5} cscθ = \frac{3}{2} secθ = - \frac{3 \sqrt{5}}{5} cotθ = - \frac{\sqrt{5}}{2}$

Step by step solution

Step 1. Given Information

Given that the function is $sinθ = \frac{2}{3}, tanθ < 0 .$ To find the exact values of $θ$ .

Step 2. Using theorem and signs

There is a difference in sign due to the quadrant of II or IV. When $y = 2 > 0,$ the quadrant is in II. The angle lies to the point $P = (x, y)$ in the standard position. The radius of the circle has $r^{2} = x^{2} + y^{2} .$ The values are $sinθ = \frac{y}{r}, y = 2, r = 3 .$ On solving and substituting,

$x^{2} = r^{2} - y^{2} x^{2} = {(3)}^{2} - {(2)}^{2} x^{2} = 9 - 4 x^{2} = 5 x = - \sqrt{5}$

Substitution to find exact values

On substituting the values for exact values are,

$cosθ = \frac{x}{r} cosθ = - \frac{\sqrt{5}}{3} tanθ = \frac{y}{x} tanθ = - \frac{2}{\sqrt{5}} . \frac{\sqrt{5}}{\sqrt{5}} tanθ = - \frac{2 \sqrt{5}}{5}$

Substitution to find exact values

On finding the exact values are,

$cscθ = \frac{r}{y} cscθ = \frac{3}{2} secθ = \frac{r}{x} secθ = - \frac{3}{\sqrt{5}} . \frac{\sqrt{5}}{\sqrt{5}} secθ = - \frac{3 \sqrt{5}}{5} cotθ = \frac{x}{y} cotθ = - \frac{\sqrt{5}}{2}$

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Find the exact value of each of the remaining trigonometric functions of $θ .$
role="math" localid="1646521470980" $sinθ = \frac{2}{3}, tanθ < 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Using theorem and signs

Substitution to find exact values

Substitution to find exact values

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Find the exact value of each of the remaining trigonometric functions of θ.role="math" localid="1646521470980" sinθ=23,tanθ<0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Using theorem and signs

Substitution to find exact values

Substitution to find exact values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Logic and Functions

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Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find the exact value of each of the remaining trigonometric functions of $θ .$
role="math" localid="1646521470980" $sinθ = \frac{2}{3}, tanθ < 0$