Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. In this context, knowing the cosine value allows us to find the sine and tangent values using the Pythagorean identity, which is essential for calculating tan 2θ.
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Introduction to Trigonometric Functions
Double Angle Formulas
Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For tangent, the formula is tan 2θ = 2tan θ / (1 - tan² θ). This formula is crucial for finding tan 2θ once we determine tan θ from the given cosine value.
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Quadrants and Angle Signs
Understanding the unit circle and the signs of trigonometric functions in different quadrants is vital. Since θ is in quadrant IV, cosine is positive while sine and tangent are negative. This knowledge helps in accurately determining the values of sine and tangent, which are necessary for calculating tan 2θ.
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