Sum and Difference Identities of Sine and Cosine

TL;DR

Learn how to use sum and difference formulas to evaluate trigonometric expressions and find exact values.

Transcript

now let's talk about the sum and difference formulas that you need to know the first one is this one cosine alpha plus or minus beta is equal to cosine alpha times cosine beta minus plus sine alpha sine beta now this equation is a way to write two equations let's say if we have alpha plus beta you need to use the negative sign so whatever sign is h... Read More

Key Insights

• 😑 The sum and difference formulas for cosine, sine, and tangent are essential in evaluating trigonometric expressions.
• ➕ The sign on top of the formula determines whether the sign in the formula should be minus or plus.
• 🔺 Common angles on the unit circle can be used to find angles that add up or subtract to a given angle.

Questions & Answers

Q: What are the sum and difference formulas for cosine, sine, and tangent?

The sum and difference formulas for cosine are cosine alpha plus or minus beta equals cosine alpha times cosine beta minus plus sine alpha sine beta. For sine, it is sine alpha plus or minus beta equals sine alpha cosine beta plus or minus cosine alpha sine beta. And for tangent, it is tangent alpha plus or minus beta equals tangent alpha plus or minus tan beta divided by one minus plus tan alpha tan beta.

Q: How can we use the sum and difference formulas to evaluate the sine of 15 degrees?

To evaluate the sine of 15 degrees, we can use the formula sine alpha minus beta equals sine alpha cosine beta minus cosine alpha sine beta. By choosing alpha as 45 degrees and beta as 30 degrees, we can substitute these values into the formula and simplify to get the exact value of sine 15 degrees.

Q: Can you provide an example of using the sum and difference formulas to evaluate the exact value of sine 105 degrees?

Sure! By choosing alpha as 60 degrees and beta as 45 degrees, we can use the formula sine alpha plus beta equals sine alpha cosine beta plus cosine alpha sine beta. Substituting these values and simplifying, we can find the exact value of sine 105 degrees.

Q: How do we evaluate the expression cosine 2 pi/3 - pi/6 using the sum and difference formulas?

We can evaluate cosine 2 pi/3 - pi/6 by using the formula cosine alpha minus beta equals cosine alpha cosine beta plus sine alpha sine beta. By choosing alpha as 2 pi/3 and beta as pi/6, we can substitute these values into the formula and simplify to find the value of the expression.

Summary & Key Takeaways

• The sum and difference formulas for cosine, sine, and tangent are explained.

• Examples are provided to demonstrate how to use the formulas to evaluate trigonometric expressions.

• The value of sine, cosine, and tangent of specific angles are calculated using the sum and difference formulas.