Given the expression sin(3x+10°)=cos(2x−10°) , which of the following uses the relationship between the sine and cosine of complementary angles to correctly determine the value of x?
The relationship between the sine and cosine of complementary angles states that for any angle A, sin(A) = cos(90° - A) and cos(A) = sin(90° - A).
In the given expression, sin(3x+10°) = cos(2x - 10°), we can see that the angles inside the sin and cos functions are complementary.
Therefore, we can write:
sin(3x+10°) = sin(90° - (2x - 10°))
Since the sine function is equal, we can equate the angles:
3x + 10° = 90° - (2x - 10°)
Simplifying,
3x + 10° = 90° - 2x + 10°
Combining like terms,
3x + 2x = 90° - 10° - 10°
5x = 70°
Dividing both sides by 5,
x = 14°
Therefore, the correct value of x is 14°.