Answer :
To find the value of [tex]\(x\)[/tex] where [tex]\(\cos(x) = \sin(14^\circ)\)[/tex] and [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use a trigonometric identity known as the co-function identity. Let's break down the steps to solve this problem.
1. Use the Co-function Identity:
The co-function identity for sine and cosine states that:
[tex]\[ \sin(\theta) = \cos(90^\circ - \theta) \][/tex]
This tells us that sine and cosine are co-functions of each other, meaning the sine of an angle is equal to the cosine of its complement.
2. Set Up the Equation:
Given [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], we can use the identity:
[tex]\[ \sin(14^\circ) = \cos(90^\circ - 14^\circ) \][/tex]
Therefore:
[tex]\[ \cos(x) = \cos(90^\circ - 14^\circ) \][/tex]
3. Equate the Angles:
Since the cosine function is equal for equal angles (in the range of 0° to 90° for our purposes), this means:
[tex]\[ x = 90^\circ - 14^\circ \][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = 90^\circ - 14^\circ \][/tex]
[tex]\[ x = 76^\circ \][/tex]
5. Verify the Range:
Check if [tex]\(x\)[/tex] falls within the given range [tex]\(0^\circ < x < 90^\circ\)[/tex]:
[tex]\[ 0^\circ < 76^\circ < 90^\circ \][/tex]
This condition is satisfied.
Therefore, the value of [tex]\(x\)[/tex] is [tex]\(76^\circ\)[/tex]. Thus, the correct answer is:
A. [tex]\(76^\circ\)[/tex]
1. Use the Co-function Identity:
The co-function identity for sine and cosine states that:
[tex]\[ \sin(\theta) = \cos(90^\circ - \theta) \][/tex]
This tells us that sine and cosine are co-functions of each other, meaning the sine of an angle is equal to the cosine of its complement.
2. Set Up the Equation:
Given [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], we can use the identity:
[tex]\[ \sin(14^\circ) = \cos(90^\circ - 14^\circ) \][/tex]
Therefore:
[tex]\[ \cos(x) = \cos(90^\circ - 14^\circ) \][/tex]
3. Equate the Angles:
Since the cosine function is equal for equal angles (in the range of 0° to 90° for our purposes), this means:
[tex]\[ x = 90^\circ - 14^\circ \][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = 90^\circ - 14^\circ \][/tex]
[tex]\[ x = 76^\circ \][/tex]
5. Verify the Range:
Check if [tex]\(x\)[/tex] falls within the given range [tex]\(0^\circ < x < 90^\circ\)[/tex]:
[tex]\[ 0^\circ < 76^\circ < 90^\circ \][/tex]
This condition is satisfied.
Therefore, the value of [tex]\(x\)[/tex] is [tex]\(76^\circ\)[/tex]. Thus, the correct answer is:
A. [tex]\(76^\circ\)[/tex]
