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Find all solutions of each equation. cos x = ﹣1/2
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 3, Problema 15
In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°
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Recognize that 195° can be expressed as a sum or difference of angles whose sine and cosine values are known. For example, 195° = 180° + 15° or 195° = 210° - 15°.
Choose the difference formula for sine since 195° = 210° - 15°, and both 210° and 15° are angles with known sine and cosine values.
Recall the sine difference formula: \(\sin(A - B) = \sin A \cos B - \cos A \sin B\).
Substitute \(A = 210^\circ\) and \(B = 15^\circ\) into the formula: \(\sin 195^\circ = \sin 210^\circ \cos 15^\circ - \cos 210^\circ \sin 15^\circ\).
Use known exact values for \(\sin 210^\circ\), \(\cos 210^\circ\), \(\sin 15^\circ\), and \(\cos 15^\circ\) to express the exact value of \(\sin 195^\circ\).
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Sum and Difference Formulas for Sine
These formulas express the sine of a sum or difference of two angles in terms of the sines and cosines of the individual angles. Specifically, sin(a ± b) = sin a cos b ± cos a sin b. This allows the evaluation of sine for angles not commonly found on the unit circle by breaking them into sums or differences of known angles.
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Percorso guidato
2:25
Verifying Identities with Sum and Difference Formulas
Reference Angles and Angle Decomposition
To apply sum or difference formulas effectively, the given angle (195°) is decomposed into the sum or difference of angles with known sine and cosine values, such as 180° + 15° or 210° - 15°. Understanding how to choose these angles simplifies the calculation and helps find exact trigonometric values.
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Percorso guidato
5:31Reference Angles on the Unit Circle
Exact Values of Sine and Cosine for Special Angles
Knowing the exact sine and cosine values for special angles like 0°, 30°, 45°, 60°, 90°, and their multiples is essential. These values serve as the building blocks when using sum or difference formulas, enabling precise computation without relying on decimal approximations.
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Percorso guidato
5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
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