Section 8-4 Relationships Among the Functions

Recall…

If P is a point on the unit circle, then x = cos Θ and y = sin Θ. From the symmetry of a circle, the following relationships are true. sin (-Θ) = -y = - sin Θ cos (-Θ) = x = cos Θ Using the reciprocal relationships, we can find negative relationships for the other functions.

Relationships with Negatives

Pythagorean Relationships We can find other relationships using the unit circle. Since P(cos Θ, sin Θ) is on the unit circle,

Pythagorean Relationships We can find other relationships using algebra. What other identities can we get from:

Pythagorean Relationships What other identities can we get from:

Pythagorean Relationships What other identities can we get from:

Cofunctions The sine and cosine functions are called cofunctions as are the tangent and cotangent and the secant and cosecant. There is a special relationship between a function and its cofunction. Complete activity 1 p. 318.

Cofunctions In general, function of Θ = cofunction of the complement of Θ.

Identities Each of the trigonometric relationships given is true for all values of the variable for which each side of the equation is defined. Such relationships are called trigonometric identities, just as is called an algebraic identity.

Proving Identities… some suggestions Learn well the formulas given above (or at least, know how to find them quickly). The better you know the basic identities, the easier it will be to recognize what is going on in the problems. Work on the most complex side and simplify it so that it has the same form as the simplest side.

Proving Identities… some suggestions Don't assume the identity to prove the identity. This means don't work on both sides of the equals side and try to meet in the middle. Start on one side and make it look like the other side. Many of these come out quite easily if you express everything on the most complex side in terms of sine and cosine only.

Proving Identities… some suggestions In most examples where you see power 2 (that is, 2), it will involve using the identity sin2 θ + cos2 θ = 1 (or one of the other 2 formulas that we derived above). Using these suggestions, you can simplify and prove expressions involving trigonometric identities.

Activity Complete activity 2 p. 319

Example Since Then And

Example Since Then And

Example Since Then And

Example Simplify. Sin Θ sec Θ cot Θ 1 + csc x(

Example Prove: