Use identities to fill in each blank with the appropriate trigonometric function name.
tan 24° = 1/ _____ 66°

검증된 단계별 안내

Recall the complementary angle identity for tangent: \(\tan(\theta) = \cot(90^\circ - \theta)\), which means tangent of an angle is the cotangent of its complement.

Notice that \(24^\circ\) and \(66^\circ\) are complementary angles because \(24^\circ + 66^\circ = 90^\circ\).

Rewrite \(\tan 24^\circ\) using the complementary angle identity: \(\tan 24^\circ = \cot 66^\circ\).

Recall that cotangent is the reciprocal of tangent: \(\cot \alpha = \frac{1}{\tan \alpha}\).

Therefore, \(\tan 24^\circ = \frac{1}{\tan 66^\circ}\), so the blank should be filled with the function name 'tan'.

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주요 개념

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Complementary Angles in Trigonometry

Complementary angles are two angles whose measures add up to 90°. In trigonometry, the functions of complementary angles are related, such as sin(θ) = cos(90° - θ). This relationship helps in transforming trigonometric expressions involving complementary angles.

Reciprocal Trigonometric Functions

Reciprocal functions are pairs of trigonometric functions where one is the reciprocal of the other, like tan(θ) and cot(θ). Specifically, cotangent is the reciprocal of tangent, meaning cot(θ) = 1/tan(θ). Recognizing these pairs is essential for rewriting expressions.

Using Trigonometric Identities to Simplify Expressions

Trigonometric identities are equations involving trig functions that hold true for all angle values. They allow substitution and simplification of expressions, such as using tan(θ) = 1/cot(θ) or relating functions of complementary angles to rewrite and solve problems efficiently.

Use identities to fill in each blank with the appropriate trigonometric function name.tan 24° = 1/ _____ 66°