Sum and Difference Identities for Cosine

Understanding Cosine Sum and Difference Identities 00:05

"The cosine of a sum or difference can be expressed with specific identities."

• The video introduces the cosine sum and difference identities, which are essential for calculating function values involving cosine.

• The identity states that the cosine of A plus B equals the cosine of A times the cosine of B minus the sine of A times the sine of B. Conversely, the cosine of A minus B equates to the cosine of A times the cosine of B plus the sine of A times the sine of B.

• It is noted that these two identities can also be represented in a cleaner form, where the addition or subtraction in one identity implies a flip in the sign of the other identity.

Proof of the Identities 00:55

"The proof of these identities is straightforward if you follow the diagram."

• A proof for these cosine identities is provided in the video, encouraging viewers to pause and understand the reasoning behind them by analyzing the relevant diagram.

• The method involved calls for viewers to follow specific instructions to effectively grasp how these identities are derived.

Application through Example Problems 01:18

"Let’s find the cosine of the sum of two angles using these identities."

• The presenter demonstrates how to apply these identities by solving examples, starting with sine values for angles A and B. For angle A, sine equals 12/13 in the second quadrant and for angle B, sine equals 4/5 in the first quadrant.

• The goal is to calculate the cosine of A plus B by correctly using the identities, which involves first determining the cosine values for A and B from their sine values and standard angle positions.

Step-by-Step Calculation of Cosine Values 01:46

"We can find cosine A and B by sketching the angles in standard position."

• The video illustrates how to find cosine A and cosine B by sketching these angles. For angle A, the adjacent side and hypotenuse are derived, leading to cosine A being -5/13.

• For angle B, cosine B is found to be 3/5, thus completing the necessary components to utilize the identity for calculating cosine A plus B.

Result of the Calculation 03:17

"The calculated cosine of A plus B turns out to be -63 over 65."

• The calculations culminate in finding that the cosine of A plus B equals -63 over 65, demonstrating the successful use of the identities in a practical example.

Exploring Exact Values — Cosine of 15 Degrees 03:33

"We can express 15 degrees as a difference of 45 degrees and 30 degrees."

• The video then continues with another example to find the exact value of cosine of 15 degrees, leveraging sum and difference identities.

• Rather than using typical reference angles, the presenter uses the angles 45 degrees and 30 degrees to derive a functional equation that leads to cosine 15 degrees.

Final Calculation for Cosine of 15 Degrees 05:28

"The exact value of cosine 15 degrees is the sum of square root 6 plus square root 2 over 4."

• Upon completing calculations consisting of substituting known values and manipulating fractions, the conclusion is reached that cosine 15 degrees equals the sum of square root 6 plus square root 2 over 4.

Applying the Concepts to Radian Measure 05:59

"We need to convert radians to degrees to find the reference angles."

• Transitioning to advanced application, the video discusses working with radians, particularly how to express cosine of 7π/12 in terms of known angles by converting degrees to radians.

• By identifying compatible angles (60 degrees and 45 degrees), the same identities are employed to find the cosine value in radians.

Additional Problem Solving 08:00

"This problem fits our identity when we express it in a different form."

• Another problem is presented where combining sine and cosine terms ultimately gives a clear path to utilizing the identities to find a sum of angles leading to cosine 90 degrees.

• The identity yields a final result of zero, underlining the concept that cosine of 90 degrees is indeed zero within the unit circle.