Textbook Question
Find values of the sine and cosine functions for each angle measure.
2x, given tan x = 5/3 and sin x < 0
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6. Trigonometric Identities and More Equations
Double Angle Identities
2:36 minutesProblem 2
Textbook Question
Match each expression in Column I with its value in Column II.
(2 tan 15°)/(1 - tan² 15°)
Verified step by step guidance1
Recognize that the given expression \( \frac{2 \tan 15^\circ}{1 - \tan^2 15^\circ} \) matches the double-angle formula for tangent, which is \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).
Identify \( \theta = 15^\circ \) in the expression, so the expression simplifies to \( \tan 2 \times 15^\circ = \tan 30^\circ \).
Recall the exact value of \( \tan 30^\circ \), which is a well-known special angle in trigonometry.
Use this value to match the expression from Column I with the corresponding value in Column II.
Confirm the match by verifying the angle and the formula used, ensuring the correct pairing.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Double-Angle Formula
The tangent double-angle formula states that tan(2θ) = (2 tan θ) / (1 - tan² θ). This identity allows us to express the tangent of twice an angle in terms of the tangent of the original angle, simplifying calculations involving angle multiples.
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Double Angle Identities
Properties of the Tangent Function
The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. It is periodic with period 180°, and understanding its behavior and values at special angles like 15° is essential for evaluating expressions.
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Angle Measurement and Conversion
Angles can be measured in degrees or radians, and recognizing common angle values such as 15°, 30°, and 45° helps in applying trigonometric identities. Accurate angle measurement is crucial for correctly using formulas and matching expressions to their values.
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