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Find all solutions of each equation. 2 cos x + √ 3 = 0
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 3, Problema 17
Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. cos(135° + 30°)
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Identify the sum identity for cosine, which is: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).
Assign the given angles to the variables: let \(A = 135^\circ\) and \(B = 30^\circ\).
Substitute the values into the sum identity formula: \(\cos(135^\circ + 30^\circ) = \cos 135^\circ \cos 30^\circ - \sin 135^\circ \sin 30^\circ\).
Recall or find the exact values of the trigonometric functions for these special angles: \(\cos 135^\circ\), \(\cos 30^\circ\), \(\sin 135^\circ\), and \(\sin 30^\circ\).
Plug these exact values into the expression and simplify step-by-step to find the exact value of \(\cos(165^\circ)\).
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Sum and Difference Identities
These identities express the sine, cosine, or tangent of a sum or difference of two angles in terms of the sines and cosines of the individual angles. For cosine, the sum identity is cos(A + B) = cos A cos B - sin A sin B, which allows exact evaluation of angles not commonly found on the unit circle.
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Percorso guidato
2:25
Verifying Identities with Sum and Difference Formulas
Exact Values of Trigonometric Functions for Special Angles
Certain angles like 30°, 45°, 60°, and their multiples have known exact sine and cosine values involving square roots. Knowing these values is essential to compute expressions like cos(135° + 30°) exactly without a calculator.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Angle Addition in Degrees
Understanding how to add angles measured in degrees and apply trigonometric identities correctly is crucial. Here, 135° + 30° equals 165°, and using the sum identity requires substituting the correct angle measures and their trigonometric values.
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Percorso guidato
3:47Coterminal Angles on the Unit Circle
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