Question

Find all solutions. $$ 2 \cos (2 t)=-\sqrt{3} $$

   Find all solutions.
$$
2 \cos (2 t)=-\sqrt{3}
$$

Precalculus: An Investigation of Functions

Precalculus: An Investigation of Functions

David Lippman 2017 Edition

Chapter 7, Problem 7

Instant Answer

Solved by Expert Kimberly Waterbury

08/28/2021

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Step 1: First, we divide both sides of the equation by 2 to isolate the cosine function: $$ \cos (2 t)=-\frac{\sqrt{3}}{2} $$ Show more…

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Find all solutions. $$ 2 \cos (2 t)=-\sqrt{3} $$

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Key Concepts

Trigonometric Equations

Trigonometric equations involve finding the angles that satisfy a given relationship based on sine, cosine, tangent, etc. Solving these equations typically requires isolating the trigonometric function, using known values from the unit circle, and then applying the periodic nature of these functions to determine the general solution.

Cosine Function

The cosine function relates an angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse, and its values are well-represented on the unit circle. It is a periodic function with a period of 2?, meaning that if an angle is a solution, then adding or subtracting multiples of 2? will yield additional solutions.

Unit Circle and Special Angles

The unit circle is a crucial tool for understanding trigonometric functions and their values at key angles. It provides the exact values of cosine and sine for common angles, which is essential when solving equations with specific trigonometric values. Recognizing these special angles helps in identifying the solutions of trigonometric equations.

General Solution for Cosine Equations

The general solution for an equation involving the cosine function takes into account its periodicity. Once a primary solution angle is identified, additional solutions are obtained by adding integer multiples of the period. This concept is important because it allows us to capture all possible angles that satisfy the cosine equation.

Double Angle Consideration

When the argument of the cosine function is a multiple of the variable, such as in a double angle, it affects the period of the function. The period of cos(2t) is different from that of cos t, and this modification must be accounted for when determining the general solution. Recognizing the impact of the double angle is essential for correctly interpreting and solving the equation.

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