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Ch. 3 - Radian Measure and The Unit Circle
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 3 - Radian Measure and The Unit CircleProblem 77
Chapter 4, Problem 77

Find the exact values of s in the given interval that satisfy the given condition.


[-2π , π) ; 3 tan² s = 1

Verified step by step guidance
1
Start by writing down the given equation: \(3 \tan^{2} s = 1\).
Isolate \(\tan^{2} s\) by dividing both sides of the equation by 3: \(\tan^{2} s = \frac{1}{3}\).
Take the square root of both sides to solve for \(\tan s\): \(\tan s = \pm \frac{1}{\sqrt{3}}\).
Recall that \(\tan s = \pm \frac{1}{\sqrt{3}}\) corresponds to specific reference angles where tangent has these values. Identify these angles within one full period of tangent, which is \(\pi\).
Find all values of \(s\) in the interval \([-2\pi, \pi)\) by adding integer multiples of \(\pi\) to the reference angles, and then select those that lie within the given interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values that satisfy the given equation within a specified interval. This often requires isolating the trigonometric function and using inverse functions or known identities to determine possible solutions.
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Properties of the Tangent Function

The tangent function, tan(s), is periodic with period π and is undefined at odd multiples of π/2. Understanding its behavior, including its range and periodicity, is essential for finding all solutions within a given interval.
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Interval Notation and Solution Sets

The interval [-2π, π) specifies the domain in which solutions must be found, including -2π but excluding π. Correctly interpreting this interval ensures that all valid solutions are identified and extraneous ones outside the range are excluded.
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Related Practice
Textbook Question

Find each exact function value. See Example 3.

sin 5π/6

7
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Textbook Question

Find the exact values of s in the given interval that satisfy the given condition.


[0 , 2π) ; cos² s = 1/2

Textbook Question

Find each exact function value. See Example 3.

tan 5π/3

2
views
Textbook Question

Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.

s = ―7.4

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Textbook Question

Find each exact function value. See Example 3.

cos 3π

Textbook Question

Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.

s = 2.5

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