Given cosθ=-(15)/(17) and 180^∘<θ<270^∘ , find the exact value of each expression. sec(θ)/(2)

Name: Problem 26, Chapter 14: Algebra 2
Uploaded: 2019-08-14T22:30:16-08:00
Duration: 1 min 52 s
Channel: Taylor Shimono
Description: Given cosθ=-(15)/(17) and 180^∘<θ<270^∘ , find the exact value of each expression. sec(θ)/(2)

step 1 So for this problem, since we're finding the secant of theta over 2, we can essentially say that

Given cosθ=-(15)/(17) and 180^∘<θ<270^∘ , find the exact...

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Key Concepts

Half-Angle Identities

These identities allow the computation of trigonometric functions of half an angle using the known trigonometric functions of the original angle. They are particularly useful for problems where an angle is given but the expression involves a half-angle, enabling the derivation of exact values through formulas like cos(?/2) = ±?((1+cos?)/2) and sin(?/2) = ±?((1?cos?)/2).

Quadrant-based Sign Determination

Understanding the quadrant in which an angle or its half lies is critical for determining the correct sign (positive or negative) of trigonometric functions. This involves knowing the sign conventions for sine, cosine, and other functions in each quadrant, which in turn ensures that the appropriate sign is assigned when applying half-angle formulas or other identities.

Reciprocal Trigonometric Functions

Reciprocal functions, such as secant, are defined as the reciprocal of the primary trigonometric functions. In this case, since sec(?/2) is defined as 1/cos(?/2), once the cosine of the half-angle is computed (taking into account the correct sign), the secant can be directly obtained by finding its reciprocal.

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