Find the Exact Value cos(115)

Problem

cos(115^∘)

Solution

1. Identify the angle and determine if it is a standard angle on the unit circle. Since 115^∘ is not a standard angle, we look for sum or difference identities using standard angles like 30^∘ 45^∘ 60^∘ or 90^∘

2. Rewrite the expression using the sum of two angles. We can express 115^∘ as 70^∘+45^∘ or 135^∘−20^∘ but these do not use only standard angles. However, if the task implies the use of a calculator or specific trigonometric identities for non-standard angles, we note that 115^∘ is in the second quadrant.

3. Apply the supplement identity cos(θ)=−cos(180^∘−θ) to relate it to an acute angle.

cos(115^∘)=−cos(180^∘−115^∘)

cos(115^∘)=−cos(65^∘)

4. Recognize that 65^∘ is also not a standard angle. Without further instructions to use specific addition formulas (like 45^∘+20^∘, the exact value is typically left in trigonometric form or evaluated using a calculator if a decimal is required. In exact radical form, this value does not have a simple representation using basic square roots like √(,2)/2

5. State the value using the cosine of a sum formula if we use 115^∘=60^∘+55^∘ or 90^∘+25^∘ which results in:

cos(90^∘+25^∘)=cos(90^∘)*cos(25^∘)−sin(90^∘)*sin(25^∘)

cos(115^∘)=0⋅cos(25^∘)−1⋅sin(25^∘)

cos(115^∘)=−sin(25^∘)

Final Answer

cos(115^∘)=−sin(25^∘)

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