If sinθ=(1)/(3), θin quadrant II, find the exact value of: (a) cosθ(b) sin(θ+(π)/(6)) (c) cos(θ-

If sinθ=(1)/(3), θin quadrant II, find the exact value of:...

Key Concepts

Trigonometric Functions in Different Quadrants

This concept addresses the fact that the values (and signs) of trigonometric functions such as sine, cosine, and tangent depend on the quadrant in which the angle is located. In various quadrants of the coordinate system, although the magnitude may be determined through identities, the sign (positive or negative) of these functions must be adjusted according to the angle’s position.

Pythagorean Identity

The Pythagorean Identity, which states that sin²? + cos²? = 1, is used to compute one trigonometric function when another is known. This fundamental relationship, derived from the geometry of the unit circle, is essential for determining missing trigonometric values while ensuring that proper sign conventions based on the angle’s quadrant are maintained.

Angle Sum and Difference Identities

Angle sum and difference identities allow the evaluation of the sine, cosine, and tangent of sums or differences of angles. These formulas express trigonometric functions of compound angles as combinations of the functions of the individual angles, making it possible to break down complex expressions into simpler, known components.

Exact Trigonometric Values

Exact trigonometric values are those expressed in a precise form involving rational numbers and radicals, rather than decimal approximations. These values are typically derived from the unit circle or special triangles and are crucial when problems require exact answers, ensuring precision in trigonometric computations.

Transcript

00:01 In this problem, we're looking at when you're given the exact value of a trig function, how you can use that to find other trig functions included, added, and subtracted angles.

00:13 So we're looking at sign of 1 over 3, and sign is opposite over hypotenuse.

00:21 So opposite is the side opposite from the angle theta.

00:25 Hypotenuse is the long diagonal side, and adjacent is the side left.

00:30 So opposite here is 1, hypotenuse is 3, but we still need to find side a.

00:37 So we're going to use pythagorean theorem, which is a squared plus b squared, equals c squared.

00:45 So we're looking for a plus b, it's going to be 1, and c will always be the hypothesis, so that's 3 squared.

00:54 So let me solve this out, the a squared equals 9 minus 1.

00:58 So, a equals the square root of 8, where we'll simplify it as 2 root 2.

01:04 However, in this problem, we are told that we're looking at quadrant 2.

01:10 And i like to use the mnemonic, all students take calc.

01:17 So, all trig functions are positive in the first quadrant, sign is positive in the second quadrant, tangents positive in the third, and cosine is positive in the fourth.

01:29 So that means with this problem, adjacent needs to be negative because cosine and tangent will be negative.

01:38 So the first thing we're asked is cosine of theta.

01:43 So with that, it's adjacent.

01:45 So negative 2 root 2 over hypotenuse 3.

01:52 Then we're asked sine of theta plus pi over 6.

02:00 And that expands out to sine theta, cosine pi over 6.

02:06 And with sign, the sign there is the same as the sign here, so plus sign of pi over 6, cosine theta.

02:19 So we're going to have 1 3 times root 3 over 2 plus 1 half times negative 2 root 2 over 3.

02:36 So if we look, we can see both of these should have a denominator of 6.

02:43 So we'll go ahead and add them together.

02:49 And this is our final answer there.

02:54 Next, they ask us to find cosine of theta minus pi over 3...