In a right triangle, the acute angles have the relationship sin(x+12°)=cos(18°+2x) . What is the value of x? What is the measure of the smaller angle? Use 1–2 sentences to explain how you arrived at your answers.
To solve the equation sin(x+12°)=cos(18°+2x), we can use the identity sin(x+12°) = cos(90° - x - 12°). Simplifying the equation gives cos(90° - x - 12°) = cos(18° + 2x). Since the two sides of the equation are equal, their angles must be equal as well. Therefore, we have 90° - x - 12° = 18° + 2x. We can then solve for x. Once we find the value of x, we can substitute it into either angle expression to find the measure of the smaller angle.
what is the value of X
To find the value of x, we can solve the equation 90° - x - 12° = 18° + 2x. Simplifying the equation gives 78° - x = 18° + 2x. Rearranging the equation gives 3x = 60°, and solving for x gives x = 20°. Therefore, the value of x is 20°.
plug 20 in the equation
If we plug x = 20° into the equation sin(x+12°)=cos(18°+2x), we have sin(20° + 12°) = cos(18° + 2(20°)). Simplifying this gives sin(32°) = cos(58°). We can see that both the sine and cosine functions are positive in the first quadrant, so the equation holds true.
what is the measure of the smaller angle
The measure of the smaller angle can be found by substituting x = 20° into one of the angle expressions. Let's use the expression x + 12°. Plugging in x = 20° gives us 20° + 12° = 32°. Therefore, the measure of the smaller angle is 32°.
