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Chapter 4: Problem 65

Find all real numbers in the interval \([0,2 \pi)\) that satisfy each equation. Round approximate answers to the nearest tenth. $$ 4 \cdot 16^{\sin ^{2}(x)}=64^{\sin (x)} $$

Short Answer

Expert verified
The solutions are \( x = \pi/2, \pi/6, \ and 5 \pi/6 \)

Step by step solution

01

Simplify the Equation

First, simplify the given equation. Recall that 16 can be expressed as 4 squared \(4^2 = 16\) and 64 can be expressed as 4 cubed \(4^3 = 64\). Substitute these into the equation: \[ 4 \times (4^2)^{\sin^2(x)} = (4^3)^{\sin(x)} \]
02

Apply Exponent Rules

Next, apply the power rule \[ (a^m)^n = a^{m \times n} \] to the equation: \[ 4 \times 4^{2 \sin^2(x)} = 4^{3 \sin(x)} \]
03

Combine the Exponents

Express the left side of the equation as a single exponent: \[ 4^1 \times 4^{2 \sin^2(x)} = 4^{1 + 2 \sin^2(x)} \] Now the equation becomes: \[ 4^{1 + 2 \sin^2(x)} = 4^{3 \sin(x)} \]
04

Equate Exponents

Since the bases are the same (4), equate the exponents: \[ 1 + 2 \sin^2(x) = 3 \sin(x) \]
05

Rearrange the Equation

Rearrange the equation into standard quadratic form: \[ 2 \sin^2(x) - 3 \sin(x) + 1 = 0 \]
06

Solve the Quadratic Equation

Use the quadratic formula \[ \sin(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = -3 \), and \( c = 1 \): \[ \sin(x) = \frac{3 \pm \sqrt{9 - 8}}{4} = \frac{3 \pm 1}{4} \] The solutions are \( \sin(x) = 1 \) and \( \sin(x) = \frac{1}{2} \)
07

Find the Angles

Determine the angles corresponding to \( \sin(x) = 1 \) and \( \sin(x) = \frac{1}{2} \) in the interval \([0, 2 \pi) \): \[ x = \pi/2 \ for \ \sin(x) = 1 \] and \[ x = \pi/6, 5 \pi/6 \ for \ \sin(x) = \frac{1}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic equations
In this exercise, simplifying the trigonometric equation brought us to a quadratic equation of the form \(\text{2sin}^2(x) - 3sin(x) + 1 = 0\).

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Most popular questions from this chapter

Find the exact value of each composition without using a calculator or table. $$ \sin ^{-1}(\sin (3 \pi / 4)) $$

Find all real numbers that satisfy each equation. Round approximate answers to 2 decimal places. $$ \frac{\sin x}{8.5}=\frac{\sin (\pi / 7)}{6.3} $$

Find the exact value of each composition without using a calculator or table. $$ \arcsin (\sin (3 \pi / 4)) $$

Simplify \(\sin (-x) \cos (-x) \tan (-x)\)

Solve each equation for the indicated variable. Solve \(x=2 \cos (3 y)\) for \(y\) where \(0 \leq y \leq \frac{\pi}{3}\)
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