Textbook Question
In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c.
sin(-t - 2𝜋) - cos(-t - 4𝜋) - tan(-t - 𝜋)
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2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
1:54 minutesProblem 26
Textbook Question
In Exercises 23–26, find the exact value of each expression. Do not use a calculator.cos 2𝜋 sec 2𝜋9 9
Verified step by step guidance1
Recognize that \( \cos 2\pi \) and \( \sec 2\pi \) are trigonometric functions evaluated at the angle \( 2\pi \).
Recall that \( \cos \theta = \frac{1}{\sec \theta} \), which means \( \sec \theta = \frac{1}{\cos \theta} \).
Evaluate \( \cos 2\pi \). Since \( 2\pi \) corresponds to a full rotation on the unit circle, \( \cos 2\pi = 1 \).
Use the identity \( \sec \theta = \frac{1}{\cos \theta} \) to find \( \sec 2\pi \). Since \( \cos 2\pi = 1 \), \( \sec 2\pi = \frac{1}{1} = 1 \).
Multiply the values: \( \cos 2\pi \cdot \sec 2\pi = 1 \cdot 1 \).
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions, such as cosine (cos) and secant (sec), are fundamental in trigonometry. The cosine function relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. The secant function is the reciprocal of cosine, defined as sec(θ) = 1/cos(θ). Understanding these functions is essential for evaluating expressions involving angles.
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Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a crucial tool in trigonometry for defining the values of trigonometric functions for all angles. The coordinates of points on the unit circle correspond to the cosine and sine values of the angle formed with the positive x-axis, allowing for easy evaluation of trigonometric expressions at key angles like 0, π/2, π, and 2π.
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Exact Values of Trigonometric Functions
Exact values of trigonometric functions refer to the specific values of these functions at certain standard angles, such as 0, π/6, π/4, π/3, and π/2. For example, cos(0) = 1 and sec(0) = 1. Knowing these exact values allows for the simplification of expressions without the need for a calculator, which is essential in problems that require precise answers.
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