Given \( \sin 58^\circ = \cos p^\circ \), find p.
Question 1 Report
Given \( \sin 58^\circ = \cos p^\circ \), find p.
Answer Details
To solve this trigonometric problem, we need to use the fact that the sine of an angle and the cosine of its complement are equal. That is, if x is an acute angle, then:
sin(x) = cos(90° - x)
Using this identity, we can rewrite the given equation:
sin(58°) = cos(p°)
cos(90° - 58°) = cos(p°) (using the identity)
cos(32°) = cos(p°) (simplifying)
Now, since the cosine function is periodic with a period of 360°, any two angles whose cosine values are equal must differ by a multiple of 360°. That is:
p° = 32° + 360°n (where n is an integer)
So, there are infinitely many possible values of p° that satisfy the equation. Some examples are:
- p° = 32° (n = 0)
- p° = 392° (n = 1)
- p° = -328° (n = -1)
Note that we can find these values by adding or subtracting multiples of 360° to the initial value of 32°, since the cosine function has the same value for an angle and its coterminal angles.
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