If cosθ=(1)/(4), θin quadrant IV, find the exact value of: (a) sinθ(b) sin(θ-(π)/(6)) (c) cos(θ+

step 1 In this question, we're given the exact value of a trig function. And we're asked to use our kno

If cosθ=(1)/(4), θin quadrant IV, find the exact value of:...

If $\cos \theta=\frac{1}{4}, \theta$ in quadrant $\mathrm{IV},$ find the exact value of:
(a) $\sin \theta$
(b) $\sin \left(\theta-\frac{\pi}{6}\right)$
(c) $\cos \left(\theta+\frac{\pi}{3}\right)$
(d) $\tan \left(\theta-\frac{\pi}{4}\right)$

Key Concepts

Pythagorean Identity

This identity, which states that sin²? + cos²? = 1, is a fundamental relation in trigonometry that allows one to compute the sine when the cosine is known, or vice versa. It is essential in problems where a single trigonometric value is given and the other must be derived.

Trigonometric Sign Conventions

Understanding the sign of trigonometric functions based on the quadrant in which an angle lies is crucial. In the unit circle, certain functions take on positive or negative values depending on the quadrant, influencing the correct selection of the square root when using the Pythagorean identity.

Trigonometric Angle Addition and Subtraction Identities

These identities, such as sin(? ± ?) = sin? cos? ± cos? sin?, cos(? ± ?) = cos? cos? ? sin? sin?, and the corresponding formulas for tangent, are key in breaking down compound angles into expressions involving the sine and cosine of individual angles. They are widely used to simplify expressions involving sums or differences of angles.

Exact Values for Special Angles

Knowledge of the exact trigonometric values for common angles (like ?/6, ?/4, ?/3, etc.) is integral to solving compound angle problems exactly. These values form the building blocks for evaluating expressions that result from applying the compound angle formulas.

Transcript

00:01 In this question, we're given the exact value of a trig function.

00:05 And we're asked to use our knowledge of trigonometry and added and subtracted angles of trig values to find other exact values.

00:19 So here we're given cosine of theta equals 1 over 4.

00:23 So cosine is adjacent over hypotenuse.

00:28 So when we're labeling a triangle, so we have angles, angle theta and across from it is opposite, the long diagonal is hypotenuse, and on the other side is adjacent.

00:39 So here, a equals 1 and h equals 4.

00:43 Now we just need to find side o.

00:46 So we're going to use pythagorean theorem, which says a squared plus b squared equals c squared.

00:53 So c will always be the hypotenuse.

00:55 So we'll have 1 squared plus b squared equals 4 squared.

01:00 So we get b squared equals 15, so b equals root 15.

01:08 However, in this problem, we're told we're in quadrant 4, and i use all students take calc to remember what values are positive in each quadrant.

01:20 So in quadrant 4, cosine is positive.

01:23 That being said, sine is not, and neither is tangent, so the opposite side must be negative.

01:29 So we have negative root 15.

01:32 We're asked to find sine of theta, which is going to be opposite over hypotenuse.

01:39 We're given negative root 15 over 4 for that.

01:46 Then we are to find sine of theta minus pi over 6.

01:52 So that's sine theta, cosine pi over 6.

02:00 Then we'll have the same sign and sine pi over 6, cosine theta...