교과서 질문
Find values of the sine and cosine functions for each angle measure.
2θ, given cos θ = -12/13 and sin θ > 0
650
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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TOC.tableOfContents
- 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
6장, 문제 6
Match each expression in Column I with its value in Column II.
(2 tan (π/3))/(1 - tan² (π/3))
검증된 단계별 안내1
Recognize that the given expression \( \frac{2 \tan\left(\frac{\pi}{3}\right)}{1 - \tan^2\left(\frac{\pi}{3}\right)} \) matches the double-angle formula for tangent, which is \( \tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).
Identify \( \theta = \frac{\pi}{3} \) in the expression, so the expression simplifies to \( \tan\left(2 \times \frac{\pi}{3}\right) = \tan\left(\frac{2\pi}{3}\right) \).
Recall or calculate the value of \( \tan\left(\frac{2\pi}{3}\right) \) by considering the unit circle or reference angles.
Use the fact that \( \frac{2\pi}{3} = \pi - \frac{\pi}{3} \), and since tangent is negative in the second quadrant, \( \tan\left(\frac{2\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) \).
Finally, substitute the known value of \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \) to find the value of the expression.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Tangent Function and Its Values
The tangent function, tan(θ), is defined as the ratio of the sine and cosine of an angle θ. Knowing exact values of tangent at special angles like π/3 (60°) is essential; for example, tan(π/3) = √3. This helps in evaluating expressions involving tangent directly.
추천 영상:
5:08
Sine, Cosine, & Tangent of 30°, 45°, & 60°
Double-Angle Formula for Tangent
The double-angle formula for tangent states that tan(2θ) = (2 tan θ) / (1 - tan² θ). This formula allows simplification of expressions involving tangent of multiple angles by relating them to the tangent of a single angle, which is crucial for matching or evaluating given expressions.
추천 영상:
05:06Double Angle Identities
Simplification of Trigonometric Expressions
Simplifying trigonometric expressions involves substituting known values and applying identities to reduce the expression to a simpler or recognizable form. This skill is necessary to match expressions with their equivalent values, especially when dealing with compound angles or rational expressions.
추천 영상:
6:36Simplifying Trig Expressions
관련 실천
교과서 질문
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
sin 300°
656
views
교과서 질문
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
sin 35°
627
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교과서 질문
Find the exact value of each expression.
sin 255°
891
views
교과서 질문
Match each expression in Column I with its equivalent expression in Column II.
(tan (π/3) - tan (π/4))/(1 + tan (π/3) tan (π/4))
756
views
교과서 질문
Match each expression in Column I with its equivalent expression in Column II.
sin 60° cos 45° - cos 60° sin 45°
830
views