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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