6.1: Verifying Trigonometric Identities Date:

Pre-Calculus

Using Fundamental Identities to Verify Other Identities:

To verify an identity, we show that ______ side of the identity can be simplified so that it is identical to the other

side. In general, start with the more __________________ side of the equation and use the fundamental identities

to transform this expression into the less complicated side of the equation. In addition, another technique is to

rewrite the more complicated side in terms of ________ and __________ and then simplify this expression.

Ex 1: Verify the identity: 𝑐𝑠𝑐𝑥𝑡𝑎𝑛𝑥 = 𝑠𝑒𝑐𝑥 Ex 2: Verify the identity: 𝑐𝑜𝑠𝑥𝑐𝑜𝑡𝑥 + 𝑠𝑖𝑛𝑥 = 𝑐𝑠𝑐𝑥 Some identities are verified by factoring to simplify a trigonometric expression. Ex 3: Verify the identity: 𝑠𝑖𝑛𝑥 − 𝑠𝑖𝑛𝑥𝑐𝑜𝑠2𝑥 = sin3 𝑥

Another simplifying technique is to separate a single-term quotient into two terms using the property:

𝑎+𝑏

𝑐=

𝑎

𝑐+

𝑏

𝑐

Ex 4: Verify the identity: 1+𝑐𝑜𝑠𝜃

𝑠𝑖𝑛𝜃= 𝑐𝑠𝑐𝜃 + 𝑐𝑜𝑡𝜃

If sums or differences of fractions with trigonometric functions appear on one side, it may be helpful to use the

____________ ______________ ______________________ (LCD) to combine the fractions into a single fraction.

Then other previously mentioned techniques will be used to finish simplifying.

Note: This method is especially useful when the other side of the identity contains only one term.

Ex 5: Verify the identity: 𝑠𝑖𝑛𝑥

1+𝑐𝑜𝑠𝑥+

1+𝑐𝑜𝑠𝑥

𝑠𝑖𝑛𝑥= 2𝑐𝑠𝑐𝑥

Homework: pg. 614 #2 – 30(e)

6.1: Verifying Trigonometric Identities (Continued) Date:

Pre-Calculus

Some identities are verified using a technique that may remind you of rationalizing the denominator (Multiplying the

conjugate).

If there is a term like 1 + 𝑠𝑖𝑛𝑥 in the denominator, multiply the numerator and denominator by _____________

(the conjugate).

Ex 1: Verify the identity: 𝑐𝑜𝑠𝑥

1+𝑠𝑖𝑛𝑥=

1−𝑠𝑖𝑛𝑥

𝑐𝑜𝑠𝑥

Remember, when all else fails, change everything into __________ and ___________ and simplify.

Ex 2: Verify the identity: 𝑠𝑒𝑐𝑥+csc(−𝑥)

𝑠𝑒𝑐𝑥𝑐𝑠𝑐𝑥= 𝑠𝑖𝑛𝑥 − 𝑐𝑜𝑠𝑥

Some identities are not easily verified by working with only one side. In these cases, you can work with each side

separately and show that both sides are equal to the same trigonometric expression.

Ex 3: Verify the identity: 1

1+𝑠𝑖𝑛𝜃+

1

1−𝑠𝑖𝑛𝜃= 2 + 2 tan2 𝜃

Guidelines for Verifying Trigonometric Identities:

o Work with each side of the equation _______________________ of the other side. Start with the more

complicated side and transform it in a step-by-step fashion until it looks exactly like the other side.

o Analyze the identity and look for opportunities to apply the fundamental identities.

o Try using one or more of the following techniques:

Rewrite the more complicated side in terms of __________ and ___________

Factor out the greatest common factor (GCF)

Separate a single-term quotient into two terms

Combine fractional expressions using the least common denominator (LCD)

Multiply the numerator and denominator by a binomial factor that appears on the other side of the

identity

o Don’t be afraid to stop and start over again if you are not getting anywhere. It is a puzzle!

Ex 4: Verify the identity: 1+𝑐𝑜𝑠𝑥

1−𝑐𝑜𝑠𝑥= (𝑐𝑠𝑐𝑥 + 𝑐𝑜𝑡𝑥)2

Ex 5: Verify the identity: (𝑐𝑜𝑠𝜃 − 𝑠𝑖𝑛𝜃)2 + (𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃)2 = 2

Ex 6: Verify the identity: (cos2 𝑥−sin2 𝑥)

1−tan2 𝑥= cos2 𝑥

Ex 7: Verify the identity: 𝑡𝑎𝑛2𝜃+𝑐𝑜𝑡2𝜃

𝑐𝑠𝑐2𝜃= 𝑠𝑒𝑐2𝜃

Ex 8: Rewrite 𝑐𝑜𝑠𝑥

1+𝑠𝑖𝑛𝑥+ 𝑡𝑎𝑛𝑥 in terms of 𝑐𝑜𝑠𝑥

Ex 9: Use the graph to make a conjecture about what the right side of the identity should be. Then prove your conjecture.

𝑐𝑜𝑠𝑥 + 𝑐𝑜𝑡𝑥𝑠𝑖𝑛𝑥

𝑐𝑜𝑡𝑥= ___________________

Ex 10: Use the graph to make a conjecture about what the right side of the identity should be. Then prove your conjecture.

1

𝑠𝑒𝑐𝑥+𝑡𝑎𝑛𝑥+

1

𝑠𝑒𝑐𝑥−𝑡𝑎𝑛𝑥= _____________

Homework: pg. 614 # pg. 614 #32 – 66(e), 67, 68

6.2: Sum and Difference Formulas Date: Pre-Calculus Sum and Difference Formulas for Cosines and Sines: cos(𝛼 + 𝛽) = 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 − 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽

cos(𝛼 − 𝛽) = 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽

sin(𝛼 + 𝛽) = 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 + 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽

sin(𝛼 − 𝛽) = 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 − 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽

Ex 1: We know that 𝑐𝑜𝑠30° = ______________. Obtain this exact value using the identity 𝑐𝑜𝑠30° = cos (90° − 60°) and the difference formula for cosines. Ex 2: Find the exact value of 𝑐𝑜𝑠70°𝑐𝑜𝑠40° + 𝑠𝑖𝑛70°𝑠𝑖𝑛40°

Ex 3: Verify the identity: cos(𝛼−𝛽)

𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽= 1 + 𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽

Ex 4: Find the exact value of sin5𝜋

12 using the fact that

5𝜋

12=

𝜋

6+

𝜋

4

Ex 5: Suppose that 𝑠𝑖𝑛𝛼 =4

5 for a quadrant II angle 𝛼 and 𝑠𝑖𝑛𝛽 =

1

2 for a quadrant I angle 𝛽. Find the exact value of

each of the following: a) 𝑐𝑜𝑠𝛼 c) cos (𝛼 + 𝛽) d) sin (𝛼 + 𝛽) b) 𝑐𝑜𝑠𝛽 Sum and Difference Formulas for Tangents

tan(𝛼 + 𝛽) =𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽

1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽

tan(𝛼 − 𝛽) =𝑡𝑎𝑛𝛼−𝑡𝑎𝑛𝛽

1+𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽

Ex 6: Verify the identity: tan(𝑥 + 𝜋) = 𝑡𝑎𝑛𝑥 Homework: pg. 623 #2 – 38(e)

6.2: Sum and Difference Formulas Continued Date: Pre-Calculus

Ex 1: Derive the identity for tan (𝛼 + 𝛽) using tan(𝛼 + 𝛽) =sin(𝛼+𝛽)

cos(𝛼+𝛽)

by dividing numerator and denominator by 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽.

Ex 2: Show that: tan (𝜃 +𝜋

4) =

𝑐𝑜𝑠𝜃+𝑠𝑖𝑛𝜃

𝑐𝑜𝑠𝜃−𝑠𝑖𝑛𝜃

Ex 3: Verify the identity: sin(𝛼+𝛽)

sin(𝛼−𝛽)=

(𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽)

𝑡𝑎𝑛𝛼−𝑡𝑎𝑛𝛽

Ex 4: Given that 𝑐𝑜𝑠𝛼 =8

17 and 𝛼 is in Quadrant IV and 𝑠𝑖𝑛𝛽 = −

1

2 and 𝛽 is in Quadrant III, find

a) cos(𝛼 + 𝛽) b) sin(𝛼 + 𝛽) c) tan(𝛼 + 𝛽) Homework: pg. 624 #40 – 68(e)

6.3: Half Angle Identity Examples (Continued) Date: Pre-Calculus Ex 1: Use 𝑐𝑜𝑠210° to find 𝑐𝑜𝑠105°

Ex 2: If 𝑡𝑎𝑛𝛼 =4

3 and 180° < 𝛼 < 270°, find:

a) sin𝛼

2 b) cos

𝛼

2 c) tan

𝛼

2

Ex 3: Verify that 𝑡𝑎𝑛𝜃 =𝑠𝑖𝑛2𝜃

1+𝑐𝑜𝑠2𝜃

Ex 4: Write sin4 𝑥 without powers of trigonometric functions greater than 1. Hint: Use the Power Reducing

Identities from the lab: sin2 𝜃 =1−𝑐𝑜𝑠2𝜃

2, cos2 𝜃 =

1+𝑐𝑜𝑠2𝜃

2

Homework: pg. 635 #36 – 54, 56, 58, 59 – 62, 70 – 78(e)

6.4: Product-to-Sum and Sum-to-Product Formulas Date: Pre-Calculus Product-to-Sum Formulas: (Use these to write products of sines and/or cosines as sum or differences)

𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 =1

2[cos(𝛼 − 𝛽) − cos(𝛼 + 𝛽)]

𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 =1

2[cos(𝛼 − 𝛽) + cos(𝛼 + 𝛽)]

𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 =1

2[sin(𝛼 + 𝛽) + sin(𝛼 − 𝛽)]

𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 =1

2[sin(𝛼 + 𝛽) − sin(𝛼 − 𝛽)]

*These formulas will be provided on tests and quizzes

Ex 1: Verify the second formula 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 =1

2[cos(𝛼 − 𝛽) + cos(𝛼 + 𝛽)]

Ex 2: Express each of the following products as a sum or difference. a) 𝑠𝑖𝑛5𝑥𝑠𝑖𝑛2𝑥 b) 𝑐𝑜𝑠7𝑥𝑐𝑜𝑠𝑥 Sum-to-Product Formulas: (Use these to write the sum or difference of sines and/or cosines as products)

𝑠𝑖𝑛𝛼 + 𝑠𝑖𝑛𝛽 = 2 sin𝛼+𝛽

2cos

𝑎−𝛽

2

𝑠𝑖𝑛𝛼 − 𝑠𝑖𝑛𝛽 = 2 sin𝛼−𝛽

2cos

𝛼+𝛽

2

𝑐𝑜𝑠𝛼 + 𝑐𝑜𝑠𝛽 = 2 cos𝛼+𝛽

2cos

𝛼−𝛽

2

𝑐𝑜𝑠𝛼 − 𝑐𝑜𝑠𝛽 = −2 sin𝛼+𝛽

2sin

𝛼−𝛽

2

*These formulas will be provided on tests and quizzes

Ex 3: Verify the third formula 𝛼 + 𝑐𝑜𝑠𝛽 = 2 cos𝛼+𝛽

2cos

𝛼−𝛽

2 . (Start with the right side and use the product-sum-

formulas). Ex 4: Express each sum as a product: a) 𝑠𝑖𝑛7𝑥 + 𝑠𝑖𝑛3𝑥 b) 𝑐𝑜𝑠3𝑥 + 𝑐𝑜𝑠2𝑥

Ex 5: Verify the identity: 𝑐𝑜𝑠3𝑥−cosx

sin3x+sinx= −𝑡𝑎𝑛𝑥

Ex 6: Find the exact value of the expression sin (𝜋

12) − sin (

5𝜋

12)

Ex 7: Verify the identity: 𝑐𝑜𝑠3𝑥−𝑐𝑜𝑠5𝑥

𝑠𝑖𝑛3𝑥+𝑠𝑖𝑛5𝑥= 𝑡𝑎𝑛𝑥

Homework: pg. 643 #6 – 12(e), 18 – 38(e), 40 – 44(e) – part (a) only, 66

3

6.5: Trigonometric Equations Date: Pre-Calculus To solve an equation containing a single trigonometric function:

o Isolate the function on one side of the equation o Solve for the variable

To find all solutions to a trigonometric function:

o Solve to find all solutions on [0,2𝜋) for sine and cosine and [0, 𝜋) for tangent o Using the period for the function, write the general answers. (Use +2𝜋𝑛 for sine and cosine and +𝜋𝑛 for

tangent where n is an integer to allow for positive and negative angles).

Ex 1: Solve the equation: 5𝑠𝑖𝑛𝑥 = 3𝑠𝑖𝑛𝑥 + √3 Equations Involving Multiple Angles:

Ex 2: Solve the equation: 𝑡𝑎𝑛2𝑥 = √3 where 0 ≤ 𝑥 < 2𝜋

Ex 3: Solve the equation: sin𝑥

3=

1

2 where 0 ≤ 𝑥 < 2𝜋

Trigonometric Equations Quadratic in Form – Set Equal to Zero and Factor

Ex 4: Solve the equation: 2 sin2 𝑥 − 3𝑠𝑖𝑛𝑥 + 1 = 0 where 0 ≤ 𝑥 < 2𝜋

Ex 5: Solve the equation: 4 cos2 𝑥 − 3 = 0 where 0 ≤ 𝑥 < 2𝜋

Ex 6: Solve the equation: 𝑠𝑖𝑛𝑥𝑡𝑎𝑛𝑥 = 𝑠𝑖𝑛𝑥 where 0 ≤ 𝑥 < 2𝜋

Using Identities to Solve a Trigonometric Equation:

Ex 7: Solve the equation: 2 sin2 𝑥 − 3𝑐𝑜𝑠𝑥 = 0 where 0 ≤ 𝑥 < 2𝜋

Ex 8: Solve the equation: 𝑐𝑜𝑠2𝑥 + 𝑠𝑖𝑛𝑥 = 0 where 0 ≤ 𝑥 < 2𝜋

Homework: pg. 656 #2 – 74(e)

6.5: Trigonometric Equations (Continued) Date:

Pre-Calculus

Ex 1: Solve the equation: 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 = −1

2 where 0 ≤ 𝑥 < 2𝜋

Ex 2: Solve the equation: 𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 = −1 where 0 ≤ 𝑥 < 2𝜋

Using a Calculator to Solve Trigonometric Equation:

Ex 3: Solve each equation, correct to four decimal places for 0 ≤ 𝑥 < 2𝜋

a) 𝑡𝑎𝑛𝑥 = 3.1044 b) 𝑠𝑖𝑛𝑥 = −0.2315

Ex 4: Solve the equation, correct to four decimal places, for 0 ≤ 𝑥 < 2𝜋: cos2 𝑥 + 5𝑐𝑜𝑠𝑥 + 3 = 0

Ex 5: Solve the equation 𝑠𝑖𝑛2𝑥𝑐𝑜𝑠𝑥 + 𝑐𝑜𝑠2𝑥𝑠𝑖𝑛𝑥 =√2

2 for 0 ≤ 𝑥 < 2𝜋

Ex 6: Solve the equation sin (𝑥 +𝜋

4) + sin (𝑥 −

𝜋

4) = 1 for 0 ≤ 𝑥 < 2𝜋

Homework: MathXL: 6.4 – 6.5 Review