Textbook Question
Use identities to correctly complete each sentence.
If cos θ = 0.8 and sin θ = 0.6, then tan(-θ) = ________________.
Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 5.42
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)
Verified step by step guidance1
Recognize that the expression \( \sin\left(\frac{3\pi}{4} - x\right) \) is a sine of a difference of two angles. We can use the sine difference identity to rewrite it.
Recall the sine difference identity: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). Here, let \( A = \frac{3\pi}{4} \) and \( B = x \).
Apply the identity: \( \sin\left(\frac{3\pi}{4} - x\right) = \sin\frac{3\pi}{4} \cdot \cos x - \cos\frac{3\pi}{4} \cdot \sin x \).
Evaluate the trigonometric functions of the constant angle \( \frac{3\pi}{4} \): \( \sin\frac{3\pi}{4} \) and \( \cos\frac{3\pi}{4} \). These are standard values on the unit circle.
Substitute these values back into the expression to write \( \sin\left(\frac{3\pi}{4} - x\right) \) entirely in terms of \( \sin x \) and \( \cos x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angle Difference Identity for Sine
The angle difference identity states that sin(A - B) = sin A cos B - cos A sin B. This formula allows you to express the sine of a difference of two angles as a combination of sines and cosines of the individual angles, which is essential for rewriting sin(3π/4 - x).
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Sum and Difference of Sine & Cosine
Exact Values of Trigonometric Functions at Special Angles
Certain angles like π/4, π/3, and π/6 have known exact sine and cosine values. For example, sin(3π/4) = √2/2 and cos(3π/4) = -√2/2. Knowing these values helps simplify expressions involving these angles without using a calculator.
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Function Notation and Variable Isolation
Expressing a trigonometric function in terms of θ or x alone means rewriting the expression so that the variable appears only inside the trigonometric functions, not combined with constants. This often involves applying identities to separate constants and variables clearly.
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Related Practice
Textbook Question
Find the remaining five trigonometric functions of θ.
csc θ = -5/2, θ in quadrant III
Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1 - 1/sec² x
Textbook Question
Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α
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Textbook Question
Verify that each equation is an identity.
(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)
Textbook Question
Find sinθ.
tan θ = -(√7)/2, sec θ > 0