For the acute angles in a right triangle, sin (4x)° = cos (3x + 13) °. What is the number of degrees in the measure of the smaller angle
The answer is 44 degrees
Why 44 do you then substitute the x in the sin(4x)°
To sin( 4(11))?
Well, since we're dealing with acute angles in a right triangle, the sum of the measures of the two acute angles is always 90 degrees. So, if one angle measures 4x degrees, the other angle would be (90 - 4x) degrees.
Now, according to the equation sin (4x)° = cos (3x + 13)°, we can rewrite it as sin (4x)° = sin (90 - (3x + 13))°.
Since sine is an odd function, sin (a)° = sin (180 - a)°. Therefore, 4x = 90 - (3x + 13).
Simplifying the equation, we get 4x = 90 - 3x - 13.
Combining like terms, we have 7x = 77.
If we divide both sides of the equation by 7, we get x = 11.
So, if x = 11, the smaller angle measures 4x = 44 degrees.
Thus, the number of degrees in the measure of the smaller angle is 44 degrees.
To find the measure of the smaller angle, we need to solve the given equation: sin(4x)° = cos(3x + 13)°.
First, let's use the trigonometric identity: sin(A) = cos(90° - A). This means that the sine of an angle is equal to the cosine of its complement.
Using this identity, we can rewrite the equation as:
cos(90° - 4x)° = cos(3x + 13)°
Since the cosine function is equal for two angles if and only if the angles differ by a multiple of 360 degrees, we have:
90° - 4x = 3x + 13 + 360° × n, where n is an integer
Simplifying the equation, we have:
90° - 4x = 3x + 13 + 360° × n
Rearranging the terms:
4x + 3x = 90° - 13 + 360° × n
Combining like terms:
7x = 77 + 360° × n
Dividing both sides by 7:
x = (77 + 360° × n) / 7
Since we're interested in the measure of the smaller angle, we substitute the value of x into the equation for its measure:
angle = 4x
angle = 4 * [(77 + 360° × n) / 7]
Simplifying further gives:
angle = (308 + 1440° × n) / 7
The measure of the smaller angle, in terms of n, is (308 + 1440° × n) / 7 degrees.
The two acute angles are complementary, so
4x + 3x+13 = 90
7x = 77
Now finish it off