(x, y) (- x, y) (- x, - y) (x, - y)

Sect 5.1 Verifying Trig identities Reciprocal Co-function Pythagorean Even/Odd Quotient

If and is in quadrant II, find each function value. S A T C If and is in quadrant II, find each function value. Negative answer. Positive answer. Positive answer. What Trig. Identity has tan and sec? What Trig. Identity has tan and cot? What Trig. Identity has tan and sin?

Write cos(x) in terms of tan(x). Secant has a relationship with both tangent and cosine. Rationalize the denominator.

Write in terms of sin(x) and cos(x), and simplify the expression so that no quotients appear.

Sect 5.2 Verifying Trig identities Guidelines to follow. 1. Work with one side of the equation at a time. It is often better to work on the most complicated. Look for opportunities to factor, add fractions, square binomials or multiply a binomial by it’s conjugate to create a monomial. 3. Look to use fundamental identities. Look to see what trig functions are in the answer. 4. Convert everything to sines and cosines 5. Always try something!

Sect 5.2 Verifying Trig identities Work on the right side first. Verify. Distribute the cosecant. Rewrite to sine and cosine. Simplify the fractions. Quotient Identity for cotangent.

Sect 5.2 Verifying Trig identities Work on the left side first. Verify. Pythagorean Identity 1 + cot2x = csc2x Rewrite to sine and cosine. Simplify the fractions by canceling . Reciprocal Identity for secant.

Sect 5.2 Verifying Trig identities Work on the left side first. Verify. Rewrite the fraction as subtraction of two fractions with the same denominators. Rewrite to sine and cosine. Simplify the fractions by multiplying by the reciprocals and cancel. Reciprocal Identity for secant and cosecant.

Sect 5.2 Verifying Trig identities Work on the left side first. Verify. Pythagorean Identity 1 + tan2x = sec2x tan2x = sec2x – 1 Rewrite to sine and cosine. Rewrite as multiplication. Cancel and Simplify.

Sect 5.2 Verifying Trig identities Work on the right side first. Two terms need to be condensed to one term. Find LCD and combine the fractions. Verify. Pythagorean Identity sin2x + cos2x = 1 cos2x = 1 – sin2x Reciprocal of cosine.

Sect 5.2 Verifying Trig identities Work on the right side first. Pythagorean Identities. Verify. sin2x + cos2x = 1 cos2x – 1 = – sin2x 1 + tan2x = sec2x Convert to cosine. Multiply.

Sect 5.2 Verifying Trig identities Work on the left side first. Try to combine the two terms into one. Verify. Convert to sine and cosine. Pythagorean Identity sin2x + cos2x = 1 Rewrite as two fractions multiplied together. Reciprocals.

Sect 5.2 Verifying Trig identities Work on the right side first. Two terms need to be condensed to one term. Verify. Convert to sine and cosine. Combine. When working with binomials, try multiplying by the conjugate to create differences of squares which will incorporate the Pythagorean Identities. Pythagorean Identity sin2x + cos2x = 1 cos2x = 1 – sin2x Cancel cosine.

Sect 5.2 Verifying Trig identities Work on the left side first. Pythagorean Identity and convert to sine and cosine. Verify. 1 + cot2x = csc2x cot2x = csc2x – 1 csc2x – 1 is Diff. of Squares. Factor. Cancel (csc x + 1) Convert to sine. Combine to one term.

Sect 5.3 Sum and Difference Formulas Using Distance Formula A – B A – B B Dist. from (cos(A-B), sin(A-B)) to (1,0) = Dist. from (cosA, sinA) to (cosB,sinB) F.O.I.L. F.O.I.L. F.O.I.L. Pythagorean Identity Pythagorean Identity Pythagorean Identity Subtract by 2. – 2 – 2 Divide by –2. The Cosine of the Difference of Two Angles

The Cosine of the Difference of Two Angles Substitute (-B) for B in the formula to make the Cosine of the Sum of Two Angle. cos (– B) = cos (B) sin (– B) = – sin (B) The Cosine of the Sum of Two Angles

To make the Sine of the Sum & Difference of Two Angles we will need the Cofunction Identities for Sine and Cosine. Start with . Substitute (-B) for B in the formula to make the Sine of the Sum of Two Angle. cos (– B) = cos (B) sin (– B) = – sin (B)

To make the Tangent of the Sum & Difference of Two Angles we will need the Quotient Identities for Tangent. This is what we need divide by all the factors. cos (A) cos (B) cos (A) cos (B) Tricky manipulation: We want this fraction to have tangents in the formula. Need to divide by the same factor in both the top and bottom to make tangents. Start with where we need to divide by cosine.

This is what we need divide by all the factors. cos (A) cos (B)

Find the exact value of . Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

Find the exact value of . Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

Suppose that for a Q2 angle and for a Q1 angle . Find the exact value of each of the following. A. B. C. D.

Find the exact value of . Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

Find the exact value of . Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

Find the exact value of . Another approach. Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

Find the exact value of .

=> Sect 5.5 Dble Angle, Power Reducing, and Half Angle Formulas Double Angle Formulas: Revise the Sum of Sin, Cos, & Tan Formulas Substitute A in for B. =>

Quadrant 4. Find given and .

Find the values of the six trigonometric functions of if and . Quadrant 2. Find the values of the six trigonometric functions of if and . Choose one of the double angle identities to find a value for sine or cosine. SOH-CAH-TOA Substitute in 4/5. Subtract by 1. Divide by -2. Square root both sides, but the answer will be positive, since we are Q2.

Verify. Work on the left side first. Convert to sine and cosine with Quotient Identity. Double angle identity. 2sin(x) cos(x) = sin(2x) Cancel Rewrite the double angle formula. 2cos2x – 1 = cos(2x) 2cos2x = 1 + cos(2x)

( ) ( ) ( ) ( ) Find an identity for Substitute Dble angle Identity. ( ) ( ) Substitute Dble angle Identity. ( ) ( ) Pythagorean Identity, rewrite with all cosines. Distribute

Product to Sum & Sum to Product Formulas How to create the Product to Sum Formulas. Add and subtract Sum and Difference formulas for Sine and Cosine.

Product to Sum Formulas Sum to Product Formulas The reason we choose these two fractions for A and B is because we need two values that add up to x and two values that subtract to be y.

Product to Sum Formulas Sum to Product Formulas

Rewrite as a sum or difference of two functions Rewrite using sums to product identity.

Half Angle Formulas The + symbol in each formula DOES NOT mean there are 2 answers, instead it indicates that you must determine the sign of the trigonometric functions based on which quadrant the half angle falls in.

Find the exact value for . S A T C Verify the identity.