Concept Check Work each problem. For what angles θ between 0° and 360° is cos θ = sin θ true?

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Recall the fundamental trigonometric identity that relates sine and cosine: \(\sin \theta = \cos \theta\).

Rewrite the equation \(\cos \theta = \sin \theta\) by dividing both sides by \(\cos \theta\), assuming \(\cos \theta \neq 0\), to get \(1 = \tan \theta\).

Express the equation in terms of tangent: \(\tan \theta = 1\).

Find the general solutions for \(\theta\) where \(\tan \theta = 1\) within the interval \(0^\circ \leq \theta < 360^\circ\). Recall that \(\tan \theta = 1\) at angles where \(\theta = 45^\circ + k \times 180^\circ\), where \(k\) is an integer.

Identify the specific angles between \(0^\circ\) and \(360^\circ\) by substituting values of \(k\) to find all solutions in the given range.

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주요 개념

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Relationship Between Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions representing ratios of sides in a right triangle or coordinates on the unit circle. Understanding how their values compare at different angles is key to solving equations like cos θ = sin θ.

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin, where angles correspond to points with coordinates (cos θ, sin θ). Knowing how to interpret angles between 0° and 360° on the unit circle helps identify where sine and cosine values are equal.

Solving Trigonometric Equations

Solving equations like cos θ = sin θ involves algebraic manipulation and using identities or geometric interpretations. Recognizing that cos θ = sin θ implies tan θ = 1 allows finding specific angle solutions within the given interval.