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If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 28
In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sin² 𝜋 + cos² 𝜋 10 10
Guida verificata passo dopo passo1
Recall the Pythagorean identity in trigonometry: \(\sin^2 x + \cos^2 x = 1\) for any angle \(x\).
Identify the angle given in the problem, which is \(\frac{\pi}{10}\).
Apply the identity directly by substituting \(x = \frac{\pi}{10}\) into the formula: \(\sin^2 \left(\frac{\pi}{10}\right) + \cos^2 \left(\frac{\pi}{10}\right)\).
Since the identity holds for all angles, the expression simplifies to 1 without needing to calculate the sine or cosine values individually.
Therefore, the value of \(\sin^2 \frac{\pi}{10} + \cos^2 \frac{\pi}{10}\) is 1.
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Pythagorean Identity
The Pythagorean identity states that for any angle θ, sin²θ + cos²θ = 1. This fundamental trigonometric identity is derived from the Pythagorean theorem and is essential for simplifying expressions involving sine and cosine squared terms.
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6:25
Pythagorean Identities
Evaluating Trigonometric Functions at Specific Angles
Understanding the values of sine and cosine at key angles, such as π (180 degrees), helps in directly substituting and simplifying expressions. For example, sin(π) = 0 and cos(π) = -1, which are critical for evaluating the given expression without a calculator.
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3:48Evaluate Composite Functions - Special Cases
Using Identities to Simplify Expressions
Applying trigonometric identities allows simplification of complex expressions into simpler forms. Instead of calculating sine and cosine values separately, identities like sin²θ + cos²θ = 1 provide a straightforward way to find the value of the expression efficiently.
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6:36Simplifying Trig Expressions
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