Find the exact values of sin(θ/ 2), cos(θ/ 2), and tan(θ/ 2) for the given conditions. secθ=-4 ;

step 1 In this problem we have given sicken theta is equals to minus 4 where theta is greater than 180

Find the exact values of sin(θ/ 2), cos(θ/ 2), and tan(θ/ 2)...

Key Concepts

Quadrantal Analysis

This concept involves determining the sign (positive or negative) of trigonometric functions based on the angle's quadrant. By knowing which quadrant the original angle lies in, one can infer the appropriate sign for both the original functions and their half-angle counterparts, ensuring that the computed values are accurate and reflect the angle's position.

Pythagorean Identity

The Pythagorean identity, which states that sin²? + cos²? = 1, is a fundamental relation used to find missing trigonometric values when one value is known. In problems where one trigonometric function is given, this identity helps in calculating the companion function, while also considering the sign based on the angle’s quadrant.

Reciprocal Trigonometric Functions

This concept involves understanding that certain trigonometric functions are reciprocals of others. For example, secant is the reciprocal of cosine. Recognizing this allows one to convert given values into the corresponding cosine values, which is especially useful when applying identities that rely on sine and cosine.

Half-Angle Identities

Half-angle identities provide formulas for calculating the sine, cosine, and tangent of half an angle in terms of the cosine (or sometimes sine) of the original angle. These identities are crucial tools in trigonometry for expressing functions at half the angle and solving equations where only the full-angle value is provided.

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