Double-Angle and Half-Angle Identities Section 5.3

Objectives Apply the half-angle and/or double angle formula to simplify an expression or evaluate an angle. Apply a power reducing formula to simplify an expression.

Double-Angle Identities

Half-Angle Identities

Power-Reducing Identities

Use a half-angle identity to find the exact value of We will use the half-angle formula for sine We need to find out what a is in order to use this formula. continued on next slide

Use a half-angle identity to find the exact value of We now replace a with in the formula to get continued on next slide

Use a half-angle identity to find the exact value of continued on next slide Now all that we have left to do is determine if the answer should be positive or negative.

Use a half-angle identity to find the exact value of continued on next slide We determine which to use based on what quadrant the original angle is in. In our case, we need to know what quadrant π/12 is in. This angle fall in quadrant I. Since the sine values in quadrant I are positive, we keep the positive answer.

If find the values of the following trigonometric functions. continued on next slide For this we will need the double angle formula for cosine In order to use this formula, we will need the cos(t) and sin(t). We can use either the Pythagorean identity or right triangles to find sin(t).

If find the values of the following trigonometric functions. continued on next slide Triangle for angle t b 7 t 9

If find the values of the following trigonometric functions. continued on next slide Triangle for angle t 7 t 9 Since angle t is in quadrant III, the sine value is negative.

If find the values of the following trigonometric functions. continued on next slide Now we fill in the values for sine and cosine.

If find the values of the following trigonometric functions. continued on next slide For this we will need the double angle formula for cosine We know the values of both sin(t) and cos(t) since we found them for the first part of the problem.

If find the values of the following trigonometric functions. continued on next slide Now we fill in the values for sine and cosine.

At this point, we can ask the question “What quadrant is the angle 2t in?” This question can be answered by looking at the signs of the sin(2t) and cos(2t). is positive and The only quadrant where both the sine value and cosine value of an angle are positive is quadrant I.

If find the values of the following trigonometric functions. continued on next slide We will use the half-angle formula for sine Since we know the value of cos(t), we can just plug that into the formula.

If find the values of the following trigonometric functions. continued on next slide

If find the values of the following trigonometric functions. continued on next slide

If find the values of the following trigonometric functions. continued on next slide Now all that we have left to do is determine if the answer should be positive or negative. In order to do this, we need to know which quadrant the angle t/2 falls in. To do this we will need to use the information we have about the angle t.

What we need is information about t/2. If we divide each piece of the inequality, we will get t/2 in the middle of the inequality and bounds for the angle on the left and right sides. If find the values of the following trigonometric functions. continued on next slide The information that we have about the angle t is

If find the values of the following trigonometric functions. continued on next slide Thus we see that the angle t/2 is in quadrant II.

If find the values of the following trigonometric functions. continued on next slide Since the angle t/2 is in quadrant II, the sine value must be positive.

If find the values of the following trigonometric functions. continued on next slide We will use the half-angle formula for sine Since we know the value of cos(t), we can just plug that into the formula.

If find the values of the following trigonometric functions. continued on next slide

If find the values of the following trigonometric functions. continued on next slide

If find the values of the following trigonometric functions. continued on next slide Now all that we have left to do is determine if the answer should be positive or negative. In order to do this, we need to know which quadrant the angle t/2 falls in. To do this we will need to use the information we have about the angle t.

What we need is information about t/2. If we divide each piece of the inequality, we will get t/2 in the middle of the inequality and bounds for the angle on the left and right sides. If find the values of the following trigonometric functions. continued on next slide The information that we have about the angle t is

If find the values of the following trigonometric functions. continued on next slide Thus we see that the angle t/2 is in quadrant II.

If find the values of the following trigonometric functions. Since the angle t/2 is in quadrant II, the cosine value must be negative.

Use the power-reducing formula to simplify the expression continued on next slide We need to use the power-reducing identity for the cosine and sine functions to do this problem. In our problem the angle a in the formula will be 7x in our problem. We also need to rewrite our problem.

Use the power-reducing formula to simplify the expression continued on next slide Now we just apply the identity to get:

Use the power-reducing formula to simplify the expression continued on next slide This is much simpler than the original expression and the power (exponent) is clearly reduced.

Use the power-reducing formula to simplify the expression Is there another way to simplify this without using a power-reducing formula? continued on next slide The answer to this question is yes. The original expression is the difference of two squares and can be factoring into Now you should notice that the expression in the second set of square brackets is the Pythagorean identity and thus is equal to 1.

Use the power-reducing formula to simplify the expression Is there another way to simplify this without using a power-reducing formula? Now you should notice that what is left is the right side of the double angle identity for cosine where the angle a is 7x. This will allow us to rewrite the expression as