Find the exact values of .a ) sin((x)/(2)), b) cos((x)/(2)), and c ) tan((x)/(2)) without solv

step 1 In this problem, we have sec x as minus 4 and x lies in the second quadrant. So we have cost of

Find the exact values of .a ) sin((x)/(2)), b)...

Find the exact values of $\left.a \text { ) } \sin \left(\frac{x}{2}\right), b\right) \cos \left(\frac{x}{2}\right),$ and $c$ ) $\tan \left(\frac{x}{2}\right)$ without solving for $x,$ when $0 \leq x \leq 360^{\circ} .$
If $\sec x=-4,$ and $x$ is in quadrant II.

Key Concepts

Quadrant and Sign Determination

Understanding the sign of trigonometric functions in different quadrants is essential for accurate calculations, especially when applying square roots in formulas like the half-angle identities. The knowledge of which functions are positive or negative in a given quadrant helps in determining the correct sign of the result, ensuring the solution aligns with the constraints of the original angle’s location in the coordinate system.

Half-Angle Formulas

Half-angle formulas provide expressions for the sine, cosine, and tangent of half an angle in terms of the cosine (or sine) of the full angle. These formulas allow one to compute sin(x/2), cos(x/2), and tan(x/2) directly from the trigonometric values of x, bypassing the need to solve for x itself. They are particularly useful in problems where an exact value is required in relation to a known trigonometric function of an angle.

Pythagorean Identity

The Pythagorean identity, sin²x + cos²x = 1, is a cornerstone of trigonometry. It provides a method to find one trigonometric function when the other is known. When given cos x, for example, you can use this identity to solve for sin x, ensuring that the correct value is determined based on the particular quadrant in which x lies.

Reciprocal Trigonometric Identities

This concept deals with the relationships between trigonometric functions and their reciprocals. For instance, knowing that sec x is the reciprocal of cos x allows you to express cos x as 1/sec x. This technique is fundamental when one of the reciprocal functions is given and you need to obtain the corresponding primary trigonometric function.

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