Maths Olympiad Prep

Track / Stage 3 / 30 of 260 #30 of 1964

Problem 30

AMC 10/12, early questions
Algebra Difficulty 3.1 Find the answer

sin⁡17∘⋅cos⁡43∘+sin⁡73∘⋅sin⁡43∘\sin 17^\circ \cdot \cos 43^\circ + \sin 73^\circ \cdot \sin 43^\circ equals \_\_\_\_\_\_.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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Official solution

Solution: sin⁡17∘⋅cos⁡43∘+sin⁡73∘⋅sin⁡43∘\sin 17^\circ \cdot \cos 43^\circ + \sin 73^\circ \cdot \sin 43^\circ
=sin⁡17∘⋅cos⁡43∘+cos⁡17∘⋅sin⁡43∘= \sin 17^\circ \cdot \cos 43^\circ + \cos 17^\circ \cdot \sin 43^\circ
=sin⁡(17∘+43∘)= \sin(17^\circ + 43^\circ)
=sin⁡60∘= \sin 60^\circ
=32= \frac{\sqrt{3}}{2}.
Therefore, the answer is: 32\boxed{\frac{\sqrt{3}}{2}}.
This can be solved using the sum-to-product formulas, the formula for the sine of the sum of two angles, and the values of the trigonometric functions for special angles.
This question mainly examines the application of sum-to-product formulas, the formula for the sine of the sum of two angles, and the values of the trigonometric functions for special angles in simplifying trigonometric expressions, focusing on the transformation of ideas, and is considered a basic question.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.