Trigonometry

To solve the problem, we need to determine the value of given that .

1. Understanding the Relationship Between and :
By definition, . Therefore, the equation can be rewritten as: Multiplying both sides by (assuming ): Taking the square root of both sides:

2. Determining for :
The secant function is defined as . To find , we first need to determine when .

Case 1:
If , then (or radians). For : Thus:

Case 2:
If , then (or radians). For : Thus: However, since the problem does not specify the quadrant, we generally consider the positive value of unless otherwise stated.

3. Conclusion:
From both cases, the value of is .

Final Answer:
The value of is .

To solve the problem, we need to determine the value of given that and .

1. Relationship Between Angles:
Since , it follows that and are complementary angles. This means:

2. Using Complementary Angle Identities:
For complementary angles, the tangent of one angle is equal to the cotangent of the other angle. Specifically:

3. Substituting the Given Value:
We are given that . Therefore:

4. Final Answer:
The value of is .

To solve the problem, we need to evaluate the trigonometric expression:

1. Recognizing the Identity:
This expression matches the cosine angle addition identity:

So,

2. Using Trigonometric Values:

Final Answer:
The value of the expression is .

To solve the problem, we are given the value of and are asked to find the value of .

1. Understanding the Identity:
By definition, the secant function is the reciprocal of the cosine function:

2. Substituting the Given Value:
We are given:

So,

Final Answer:
$ \cos \theta = \frac{\sqrt{3}}{2}