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

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Recall the sine addition formula: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\).

Identify the angles in the expression: here, \(A = 45^\circ\) and \(B = \theta\).

Apply the formula to rewrite \(\sin(45^\circ + \theta)\) as \(\sin 45^\circ \cos \theta + \cos 45^\circ \sin \theta\).

Use known exact values for \(\sin 45^\circ\) and \(\cos 45^\circ\), which are both \(\frac{\sqrt{2}}{2}\).

Substitute these values back into the expression to get \(\frac{\sqrt{2}}{2} \cos \theta + \frac{\sqrt{2}}{2} \sin \theta\).

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Angle Addition Formula for Sine

The angle addition formula for sine states that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). This allows the expression of the sine of a sum of two angles as a combination of sines and cosines of the individual angles, facilitating simplification or evaluation.

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Certain angles like 45° have known exact sine and cosine values, such as sin(45°) = cos(45°) = √2/2. Using these values helps to rewrite expressions involving these angles into simpler numeric forms or functions of the variable angle.

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Common Trig Functions For 45-45-90 Triangles

Function Notation and Variable Separation

Expressing a trigonometric function as involving only θ or x means rewriting the expression so that constants and variables are clearly separated, often by substituting known values for constants and leaving the variable angle in function form, aiding clarity and further manipulation.

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