Write each function as an expression involving functions of θ or x alone. See Example 2.
sin(45° + θ)

Verified step by step guidance

1. Recognize that the expression \( \sin(45^\circ + \theta) \) is a sum of angles, which can be expanded using the sine addition formula.
2. Recall the sine addition formula: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).
3. Substitute \( a = 45^\circ \) and \( b = \theta \) into the formula: \( \sin(45^\circ + \theta) = \sin 45^\circ \cos \theta + \cos 45^\circ \sin \theta \).
4. Use the known values: \( \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} \).
5. Substitute these values into the expression: \( \sin(45^\circ + \theta) = \frac{\sqrt{2}}{2} \cos \theta + \frac{\sqrt{2}}{2} \sin \theta \).

Key Concepts

Angle Addition Formulas

The angle addition formulas are essential trigonometric identities that express the sine, cosine, and tangent of the sum of two angles in terms of the functions of the individual angles. For example, the formula for sine states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). These formulas allow us to simplify expressions involving the sum of angles, such as sin(45° + θ).

Trigonometric Values of Special Angles

Certain angles, such as 0°, 30°, 45°, 60°, and 90°, have known sine and cosine values that are often used in trigonometric calculations. For instance, sin(45°) equals √2/2 and cos(45°) also equals √2/2. Recognizing these special angle values is crucial for simplifying expressions and solving trigonometric problems efficiently.

Function Notation and Variables

Understanding function notation is vital in trigonometry, as it allows us to express relationships between angles and their corresponding trigonometric values. In the expression sin(45° + θ), θ represents a variable angle, and recognizing how to manipulate and substitute variables in trigonometric functions is key to solving problems. This concept is foundational for expressing complex trigonometric functions in simpler forms.