Find the exact values of sin(θ/ 2), cos(θ/ 2), and tan(θ/ 2) for the given conditions. tanθ=1 ;

step 1 in this problem we have given tangent theta is equal to 1 where theta is greater than minus 180

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

Key Concepts

Half-Angle Formulas

Half-angle formulas are trigonometric identities that allow one to express the sine, cosine, and tangent of half an angle in terms of the cosine (or sometimes sine) of the original angle. They are crucial for solving problems where the angle is divided by two, as they provide a direct method to compute the exact values of functions like sin(?/2), cos(?/2), and tan(?/2) by using expressions such as sin(?/2) = ±?((1 - cos ?)/2) and cos(?/2) = ±?((1 + cos ?)/2). The correct choice of the ± sign depends on the quadrant in which the half-angle lies.

Quadrant Analysis

Quadrant analysis involves determining the sign of trigonometric functions based on the angle’s location in the coordinate plane. When applying half-angle formulas, it is essential to know the quadrant of the half-angle to choose the appropriate signs for the square root expressions. This analysis uses the periodicity and symmetry properties of the trigonometric functions, ensuring that the computed values correctly reflect the function's behavior in different intervals.

Trigonometric Identities

Trigonometric identities, such as the Pythagorean identity and tangent formulas, provide the relationships between different trigonometric functions. These identities are fundamental in transforming and simplifying expressions, especially when exact values are to be found. They form the backbone of the derivation and application of formulas like the half-angle identities, enabling the computation of functions from known ratios or values.

Exact Values of Trigonometric Functions

Understanding and determining the exact values of trigonometric functions, particularly those associated with commonly encountered angles, is essential in trigonometry. Familiarity with exact values for angles allows for precise computation using identities without resorting to numerical approximations. This concept is especially important when working with half-angles and ensuring that the results are expressed in exact radical form.

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