To solve the problem of finding the values of [tex]\( x \)[/tex] within the interval [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex] such that [tex]\(\cos x = 0.7252\)[/tex], follow these steps:
1. Identify the principal value:
The principal value of an angle [tex]\( x \)[/tex] such that [tex]\(\cos x = 0.7252\)[/tex] can be found by using the inverse cosine function. By using trigonometric tables or a calculator, you determine that:
[tex]\[
x \approx \cos^{-1}(0.7252)
\][/tex]
This gives us the principal angle, which is the angle in the first quadrant where [tex]\( \cos \)[/tex] is positive.
2. Calculate the angle in radians and degrees:
The principal value of [tex]\( x \)[/tex] is found as:
[tex]\[
x_1 \approx 44^\circ
(obtained from the inverse cosine or arc cosine of 0.7252 and then converted to degrees.)
3. Find the second solution within the interval:
Since the cosine function is positive in both the first and fourth quadrants, we need to find the corresponding angle in the fourth quadrant. The angle in the first quadrant is \( 44^\circ \), thus, the fourth quadrant angle can be found by subtracting this angle from \( 360^\circ \):
\[
x_2 = 360^\circ - 44^\circ = 316^\circ
\][/tex]
4. Summary of the solutions:
The values of [tex]\( x \)[/tex] that satisfy [tex]\(\cos x = 0.7252\)[/tex] in the interval [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex] are:
[tex]\[
x = 44^\circ \quad \text{and} \quad x = 316^\circ
\][/tex]
Thus, the values of [tex]\( x \)[/tex] to the nearest degree are [tex]\( 44^\circ \)[/tex] and [tex]\( 316^\circ \)[/tex].
