CPM Homework Help : CALC Problem 1-116

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1-116.

Calculus problems often require using one or more of the trigonometric identities to solve problems. Solve each of the following equations where $x ∈ [0, 2π]$ . Use exact values.

$\text{tan } x · \text{csc }x = 2$
Hint 1(a):
Simplify the left side of the equation to solve for $x$ .
Remember that $\text{tan}x=\frac{\text{sin}x}{\text{cos}x}$ and $\text{csc}x=\frac{1}{\text{sin}x}$ .
Steps (a):
$\frac{\text{sin}x}{\text{cos}x}\cdot\frac{1}{\text{sin}x}=2$
$\frac{1}{\text{cos}x}=2$
$\text{cos}x=\frac{1}{2}$
Hint 2(a):
Use the unit circle to solve for $x$ .
Answer (a):
$\frac{\pi }{3},\ \frac{5\pi }{3}$

$\operatorname { sin } x \cdot \operatorname { cos } x = \frac { 1 } { 4 }$
Hint (b):
See part (a).

$2 \text{ sin}^2 x -\text{cos }x - 1 = 0$
Hint (c):
See part (a).

$\text{tan }x +\text{cot }x = -2$
Hint (d):
See part (a).

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