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

Verified step by step guidance

1. Recall the fundamental trigonometric identity that relates sine and cosine: \(\sin \theta = \cos \theta\).
2. Rewrite the equation \(\cos \theta = \sin \theta\) by dividing both sides by \(\cos \theta\), assuming \(\cos \theta \neq 0\), to get \(1 = \tan \theta\).
3. Express the equation in terms of tangent: \(\tan \theta = 1\).
4. 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.
5. Identify the specific angles between \(0^\circ\) and \(360^\circ\) by substituting values of \(k\) to find all solutions in the given range.

Key Concepts

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.