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Question

Difficulty: EasyBasic Trigonometric Ratios, Special Angles, and Identities

What is the exact simplified value of sin⁡60∘+cos⁡45∘\sin 60^\circ + \cos 45^\circ?

  1. 3+22\frac{\sqrt{3} + \sqrt{2}}{2}Answer
  2. B
    52\frac{\sqrt{5}}{2}
  3. C
    3+12\frac{\sqrt{3} + 1}{2}
  4. D
    3+24\frac{\sqrt{3} + \sqrt{2}}{4}

Answer

3+22\frac{\sqrt{3} + \sqrt{2}}{2}
Substituting the exact values sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2} and cos⁡45∘=22\cos 45^\circ = \frac{\sqrt{2}}{2} gives 32+22=3+22\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} = \frac{\sqrt{3} + \sqrt{2}}{2}.

Step-by-Step Solution

1
Identify the exact trigonometric values for the special angles.
sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2} and cos⁡45∘=22\cos 45^\circ = \frac{\sqrt{2}}{2}
Standard special angle values in trigonometry.
2
Substitute the values into the given expression.
32+22\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}
Direct substitution of known ratio values.
3
Combine the fractions over the common denominator of 22.
3+22\frac{\sqrt{3} + \sqrt{2}}{2}
Adding numerators over a shared common denominator.

Key Concept

Special Angles and Exact Trigonometric Values
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Topics

Basic Trigonometric Ratios, Special Angles, and Identities

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