Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 35aBlitzer - 3rd Edition
Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions: d. sin 2α 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2
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1
Identify the double angle formula for sine: .
Given , use the Pythagorean identity to find .
Calculate by rearranging the Pythagorean identity: .
Substitute and into the double angle formula: .
Simplify the expression to find the exact value of .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Double Angle Formula for Sine
The double angle formula for sine states that sin(2α) = 2sin(α)cos(α). This formula allows us to express the sine of double an angle in terms of the sine and cosine of the original angle, which is essential for solving problems involving angles that are multiples of a given angle.
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Double Angle Identities
Understanding Sine Values
The sine function relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. Knowing the value of sin(α) is crucial for finding sin(2α), as it directly influences the calculation of both sin(α) and cos(α) in the double angle formula.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°
Range of Angles
The specified ranges for α and β indicate the quadrants in which these angles lie, affecting the signs of their sine and cosine values. For example, if 0 < α < π/2, both sin(α) and cos(α) are positive, which is important for accurately calculating sin(2α) using the double angle formula.
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Related Practice
Textbook Question
Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.
sin x cos 2x
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