Find the exact value of sin2 θ, cos2 θ, sin(θ)/(2), and cos(θ)/(2) for each of the following. co

Find the exact value of sin2 θ, cos2 θ, sin(θ)/(2), and...

Key Concepts

Double-Angle Formulas

Double-angle formulas in trigonometry provide relationships that allow you to express trigonometric functions of twice an angle in terms of the functions of the original angle. For instance, the formulas sin(2?) = 2 sin? cos? and cos(2?) = cos²? - sin²? (with alternative forms such as 2cos²? - 1 or 1 - 2sin²?) are crucial for simplifying expressions and solving equations involving trigonometric functions.

Half-Angle Formulas

Half-angle formulas enable the determination of the trigonometric functions of half an angle using the functions of the original angle. The formulas sin(?/2) = ±?((1 - cos?)/2) and cos(?/2) = ±?((1 + cos?)/2) are derived from the double-angle identities and are essential for computing exact values when the angle is halved, with the sign determined by the quadrant in which the half-angle lies.

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is a fundamental relation in trigonometry that links sine and cosine of an angle. This identity allows one to compute the missing trigonometric function when one of them is known and is used to derive various other identities, such as those for double and half angles.

Quadrant and Sign Considerations

When working with trigonometric identities, identifying the correct sign of the resulting values is critical, and this is determined by the quadrant in which the angle or half-angle lies. Based on the given conditions, evaluating the sign ensures that the exact values obtained for functions like sin(?/2) and cos(?/2) correctly reflect the angle's position in the coordinate system.

Transcript

00:01 In this problem, we're told that the cosine of theta is equal to 5 -6, and that theta is between 0 and 90, meaning that it's the first quadrant angle.

00:08 So first, we want to find the sign of 2 theta, it's double angle.

00:12 Well, remember, the formula will be 2 times the sign of theta times the cosine of theta.

00:17 Well, we know the cosine of theta, but we don't know the sign of theta.

00:20 So what i'm going to do is i'm going to draw my triangle that would be in the first quadrant, and because we're given cosine, we know the adjacent side is 5, and the opposite side is 6.

00:31 So we just need, or sorry, the hypotenuse is six.

00:35 We need to find the opposite side.

00:37 Well, we can do that using the pythagorean theorem.

00:39 So we'll have x squared plus 5 squared is equal to 6 squared.

00:43 Well, 5 squared is 25, and 6 squared is 36.

00:48 So if we subtract 25, we'll have x squared is equal to 11, so x would equal to plus or minus the square of 11.

00:56 But remember, we're in the first quadrant, so x has to be positive root 11.

01:03 Now that we know the opposite side, we now know that sine will be, sign of theta will equal to root 11 over 6.

01:12 Now we can go ahead and find the sign of 2 theta.

01:15 So when we do this, we're going to have 2 times the sine of theta, which is root 11 all over 6, times the cosine of theta, which is 5 over 6.

01:26 And now we just need to multiply.

01:28 Well, 2 times the square of 11 over 6 times the square or sorry, times just 5 over 6 is equal to 5 root 11 all over 18.

01:40 Next, we want to find the cosine of 2 theta.

01:45 So we can do that simply by using the formula 1 minus 2 sine squared theta.

01:55 Because remember, we now know that sign is root 11 over 6.

01:58 So we'll have 1 minus 2 times the quantity of root 11 over 6 quantity squared.

02:04 Well, when we square, or yeah, when we square the square of 11 over 6, we're just going to get 11 over 36.

02:11 So we have 1 minus 2 times 11 over 36.

02:15 Well, 2 times 11 over 36 is 11 over 18.

02:18 So we have 1 minus 11 over 18.

02:24 And 1 minus 11 over 18 is just 17 over 18.

02:29 Now we're next being asked to find the sign of theta root 2.

02:32 As you can see, i've ran out of room...

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