Sum and Difference Identities Section 5.2

Objectives Apply a sum or difference identity to evaluate the sine or cosine of an angle.

Sum and Difference Identities The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like

Use a sum or difference identity to find the exact value of In order to answer this question, we need to find two of the angles that we know to either add together or subtract from each other that will get us the angle π/12. Let’s start by looking at the angles that we know: continued on next slide

Use a sum or difference identity to find the exact value of We have several choices of angles that we can subtract from each other to get π/12. We will pick the smallest two such angles: Now we will use the difference formula for the sine function to calculate the exact value. continued on next slide

Use a sum or difference identity to find the exact value of For the formula a will be and b will be This will give us continued on next slide

Use a sum or difference identity to find the exact value of For the formula a will be and b will be This will give us

Simplify using a sum or difference identity In order to answer this question, we need to use the sine formula for the sum of two angles. For the formula a will be and b will be continued on next slide

Simplify using a sum or difference identity

Simplify using a sum or difference identity In order to answer this question, we need to use the cosine formula for the difference of two angles. For the formula a will be and b will be continued on next slide

Simplify using a sum or difference identity

Find the exact value of the following trigonometric functions below given and For this problem, we have two angles. We do not actually know the value of either angle, but we can draw a right triangle for each angle that will allow us to answer the questions. continued on next slide

Find the exact value of the following trigonometric functions below given and Triangle for α b 3 α 7 continued on next slide

Find the exact value of the following trigonometric functions below given and Triangle for β 4 a β 5 continued on next slide

Find the exact value of the following trigonometric functions below given and 3 α 7 Now that we have our triangles, we can use the cosine identity for the sum of two angles to complete the problem. 4 3 β 5 continued on next slide

Find the exact value of the following trigonometric functions below given and 3 α 7 Note: Since α is in quadrant Iv, the sine value will be negative Now that we have our triangles, we can use the cosine identity for the sum of two angles to complete the problem. 4 3 β 5 Note: Since β is in quadrant II, the cosine value will be negative continued on next slide

Find the exact value of the following trigonometric functions below given and 3 α 7 4 3 β 5 continued on next slide

Find the exact value of the following trigonometric functions below given and While we are here, what are the possible quadrants in which the angle α+β can fall? In order to answer this question, we need to know if cos(α+β) is positive or negative. We can type the value into the calculator to determine this. When we do this, we find that cos(α+β) is positive. The cosine if positive in quadrants I and IV. Thus α+β must be in either quadrant I or IV. We cannot narrow our answer down any further without knowing the sign of sin(α+β). continued on next slide

Find the exact value of the following trigonometric functions below given and 3 α 7 Note: Since α is in quadrant Iv, the sine value will be negative Now that we have our triangles, we can use the cosine identity for the sum of two angles to complete the problem. 4 3 β 5 Note: Since β is in quadrant II, the cosine value will be negative continued on next slide

Find the exact value of the following trigonometric functions below given and 3 α 7 4 3 β 5