Domanda del libro di testo
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan[sin⁻¹ (− 1/2)]
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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 39
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 39Capitolo 2, Problema 39
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ cos(sin⁻¹ √2/2)
Guida verificata passo dopo passo1
Recognize that the expression is \( \cos(\sin^{-1}(\frac{\sqrt{2}}{2})) \). Here, \( \sin^{-1} \) is the inverse sine function, which gives an angle \( \theta \) such that \( \sin \theta = \frac{\sqrt{2}}{2} \).
Set \( \theta = \sin^{-1}(\frac{\sqrt{2}}{2}) \), so by definition, \( \sin \theta = \frac{\sqrt{2}}{2} \). Our goal is to find \( \cos \theta \).
Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \frac{\sqrt{2}}{2} \) into this identity to find \( \cos \theta \).
Calculate \( \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left( \frac{\sqrt{2}}{2} \right)^2 \). Simplify the right side to find \( \cos^2 \theta \).
Take the square root of \( \cos^2 \theta \) to find \( \cos \theta \). Consider the range of \( \theta = \sin^{-1}(\frac{\sqrt{2}}{2}) \) to determine the correct sign (positive or negative) of \( \cos \theta \).
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Inverse Sine Function (sin⁻¹ or arcsin)
The inverse sine function, sin⁻¹(x), returns the angle whose sine is x. It is defined for x in the interval [-1, 1] and outputs angles in the range [-π/2, π/2]. Understanding this helps identify the angle corresponding to a given sine value.
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4:03
Inverse Sine
Relationship Between Sine and Cosine
Sine and cosine are co-functions related by the Pythagorean identity: sin²θ + cos²θ = 1. Knowing one trigonometric value allows you to find the other by rearranging this identity, which is essential for evaluating expressions like cos(sin⁻¹ x).
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Percorso guidato
5:05Amplitude and Reflection of Sine and Cosine
Exact Values of Special Angles
Certain angles have well-known exact sine and cosine values, such as π/4 (45°), where sin(π/4) = cos(π/4) = √2/2. Recognizing these special angles helps in finding exact trigonometric values without a calculator.
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Percorso guidato
04:3945-45-90 Triangles
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