Find the exact values of trig functions no calculators allowed!!! Warm UP Find the exact values of trig functions no calculators allowed!!! Cos(45o) Sin(30o) Cos(15o)

How to find cos(15o) Find two angles that we know from the Unit Circle to either add together or subtract from each other that will get us the angle 15o.

Use a sum or difference identity to find the exact value of

Use a sum or difference identity to find the exact value of

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: 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: continued on next slide

Use a sum or difference identity to find the exact value of Now we will use the difference formula for the sine function to calculate the exact value.

Write the expression as a single trig expression:

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.

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

Use the following to find sin2 and cos2. 5 -12 13

Ex. 1 Use the following to find sin2 and cos2. 5 13 -12

and  terminates in the first quadrant, find the exact value of sin2 and cos2. 5

Double-Angle Identities Express each in terms of a single trig function. 2 sin 0.45 cos 0.45 sin 2x = 2sin x cos x sin 2(0.45) = 2sin 0.45 cos 0.45 sin 0.9 = 2sin 0.45 cos 0.45

Using Technology Graph the function and predict the period. The period is p. Rewrite f(x) as a single trig function: f(x) = sin 2x