Which is equal to sin75°? Choices: [[cos25°][cos15°][sin25°][sin15°]]

Recognize Sum of Angles: Recognize that sin75° can be expressed as the sum of two angles for which we know the sine and cosine values. The sum of angles formula for sine is sin(a + b) = sin(a)cos(b) + cos(a)sin(b). We can express 75° as the sum of 45° and 30°, which are angles we know the sine and cosine values for.Apply Formula to sin75°: Apply the sum of angles formula to sin75°. We have sin75° = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°). We know that sin(45°) = cos(45°) = √2/2, sin(30°) = 1/2, and cos(30°) = √3/2.Substitute Known Values: Substitute the known values into the formula. This gives us sin75° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6/4) + (√2/4).Simplify Expression: Simplify the expression. We can combine the fractions to get sin75° = (√6 + √2)/4.Recognize Matching Trigonometric Function: Recognize that the value of sin75° we found must be equal to one of the trigonometric functions of the angles given in the choices. We need to find which one matches (√6 + √2)/4.Compare to Choices: Compare the value of sin75° to the choices. We know that sin(90° - x) = cos(x), so sin75° = cos(15°). Therefore, sin75° is equal to cos15°.

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Which is equal to $\sin 75^\circ$ ? $\newline$ Choices: $\newline$ (A) $\cos 25^\circ$ $\newline$ (B) $\cos 15^\circ$ $\newline$ (C) $\sin 25^\circ$ $\newline$ (D) $\sin 15^\circ$

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Math Problems

Geometry

Sine and cosine of complementary angles

Full solution

Q. Which is equal to

\sin 75^\circ

\newline

Choices:

\newline

(A)

\cos 25^\circ

\newline

(B)

\cos 15^\circ

\newline

(C)

\sin 25^\circ

\newline

(D)

\sin 15^\circ

Recognize Sum of Angles: Recognize that $\sin 75^\circ$ can be expressed as the sum of two angles for which we know the sine and cosine values. The sum of angles formula for sine is $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ . We can express $75^\circ$ as the sum of $45^\circ$ and $30^\circ$ , which are angles we know the sine and cosine values for.
Apply Formula to $\sin 75°$ : Apply the sum of angles formula to $\sin 75°$ . We have $\sin 75° = \sin(45° + 30°) = \sin(45°)\cos(30°) + \cos(45°)\sin(30°)$ . We know that $\sin(45°) = \cos(45°) = \sqrt{2}/2$ , $\sin(30°) = 1/2$ , and $\cos(30°) = \sqrt{3}/2$ .
Substitute Known Values: Substitute the known values into the formula. This gives us $\sin 75° = (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2}) = (\frac{\sqrt{6}}{4}) + (\frac{\sqrt{2}}{4})$ .
Simplify Expression: Simplify the expression. We can combine the fractions to get $\sin 75^\circ = (\sqrt{6} + \sqrt{2})/4$ .
Recognize Matching Trigonometric Function: Recognize that the value of $\sin 75^\circ$ we found must be equal to one of the trigonometric functions of the angles given in the choices. We need to find which one matches $(\sqrt{6} + \sqrt{2})/4$ .
Compare to Choices: Compare the value of $\sin 75°$ to the choices. We know that $\sin(90° - x) = \cos(x)$ , so $\sin 75° = \cos 15°$ . Therefore, $\sin 75°$ is equal to $\cos 15°$ .