Ch. 1 - Angles and the Trigonometric Functions

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Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problema 37

Capitolo 1, Problema 37

In Exercises 31–38, find a cofunction with the same value as the given expression. cos 2𝜋 5

Guida verificata passo dopo passo

Recall the cofunction identity for cosine and sine: \(\cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right)\).

Identify the given angle \(\theta\) in the expression: here, \(\theta = \frac{2\pi}{5}\).

Apply the cofunction identity by substituting \(\theta\) into the formula: \(\cos\left(\frac{2\pi}{5}\right) = \sin\left(\frac{\pi}{2} - \frac{2\pi}{5}\right)\).

Simplify the expression inside the sine function by finding a common denominator: \(\frac{\pi}{2} = \frac{5\pi}{10}\) and \(\frac{2\pi}{5} = \frac{4\pi}{10}\), so \(\frac{\pi}{2} - \frac{2\pi}{5} = \frac{5\pi}{10} - \frac{4\pi}{10} = \frac{\pi}{10}\).

Write the final cofunction expression: \(\cos\left(\frac{2\pi}{5}\right) = \sin\left(\frac{\pi}{10}\right)\).

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Cofunction Identity

Cofunction identities relate the trigonometric functions of complementary angles, such as sin(θ) = cos(90° - θ) or sin(θ) = cos(π/2 - θ) in radians. These identities allow us to express one trigonometric function in terms of another by using the complementary angle.

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Cofunction Identities

Radian Measure

Radian measure is a way to express angles based on the radius of a circle, where 2π radians equal 360 degrees. Understanding how to convert between radians and degrees or interpret angles in radians is essential for applying trigonometric identities correctly.

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Converting between Degrees & Radians

Cosine Function

The cosine function gives the x-coordinate of a point on the unit circle corresponding to a given angle. Knowing its properties, such as periodicity and symmetry, helps in identifying equivalent expressions and applying cofunction identities effectively.

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Graph of Sine and Cosine Function

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