Unit R Student Success Sheet (SSS) Trigonometric Identities Part 2 (section 5.4)

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1 Unit R Student Success Sheet (SSS) Trigonometric Identities Part 2 (section 5.4) Segerstrom High School Standards: Trig 10.0 Math Analysis Honors Name: Period: Mrs. Kirch: All Mornings 7-8am + after school on most days (just ask!) Ms. Tamaoki: Tuesday & Thursday mornings 7:30-8am Edmodo.com (all answer keys, videos, & resources are under library tab in shared folders. Math Analysis group code to join is xd1wmo Pre-Calculus for Dummies and Pre-Calculus demystified available for checkout from Mrs. Kirch Reminders: Homework is completed in spiral bound notebook only. Homework not done in homework notebook will not be accepted. All pages in homework notebook should be labeled accordingly: Unit Concept - (title of assignment) Examples: Unit R Concept 1 Practice Quiz Unit R Concept 1 Quiz Review Unit R Concept 1-4 Practice Test Success has a simple formula: do your best, and people may like it. Sam Ewing Concept # What we will be learning 1 Exact value of sums or differences (might have to find what it adds or subtracts to) 2 Using sum and difference formulas when given values (right triangles) 3 Trig of an inverse trig function (use sum and difference to prove) 4 Simplifying sum and difference identities 5 Solving trig equations with sum and difference We will be continuing our study of trigonometric identities, focusing on sum and difference identities. These identities are most helpful to us when we want to find exact values of angles that AREN T on the unit circle, but can be created by either adding or subtracting two angles that ARE on the unit circle! So, this expands our ability to use the unit circle for just 0,30,45,60, and 90 degree reference angles to many more!

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3 Exact value of sums or differences (might have to find what it adds or subtracts to) Sometimes we want to find the exact values of angles that AREN T on the UNIT CIRCLE. This concept will show us that we can find the exact value of any angle of which angles on the unit circle can ADD (sum) or SUBTRACT (difference) to that angle! We must first realize what this DOESN T mean For the following problems, find the exact values (this means your answer will have RADICALS in it and NOT DECIMALS!) of the sine, cosine, and tangent of the angle. We must first decide what angles to use to either add or subtract to the one we want. There is often MORE THAN ONE OPTION for this! o = 60 o + 45 o U ordered pair: V ordered pair: Sin Cos tan

4 Sin 2. 75o = Cos Tan U ordered pair: V ordered pair: 3. = U ordered pair: V ordered pair: o= U ordered pair: V ordered pair:

5 Sin 5. = Cos Tan U ordered pair: V ordered pair: Write the expression as either the sine, cosine, or tangent of an angle 6. cos60ocos20o sin60osin20o Additional Problems on extra videos Extra video #1 covers the following problems. Use these as extra practice and the videos as a guide if you need help. 9. Sin(405o+120o) vs. Sin405o + sin 120o 10. Sin( ) vs. Sin( ) Extra video #2 covers the following problems. Use these as extra practice and the videos as a guide if you need help. Find the sine, cosine, and tangent of the following angles o = 135o + 30o o 13. PQ problems: Find the sine, cosine, and tangent of the following angles Write the expression as either the sine, cosine, or tangent of an angle 20. sin110ocos80o +cos110osin80o (these are in radians!) 21. sin3.5cos1.2 cos3.5sin

6 PT problems: Find the sine, cosine, and tangent of the following angles Write the expression as either the sine, cosine, or tangent of an angle 28. (these are in radians!) Cos(.96)cos(.42) + sin(.96)sin(.42) Using sum and difference formulas when given values (right triangles) Begin by : Drawing the appropriate triangles in the correct quadrants. Find the missing piece of the triangle. BE CAREFUL OF THE SIGNS (left and down is negative, right and up is positive!) Plug into the sum and difference formulas. 1. Find the exact value of the trigonometric function given that and. (Both u and v are in Quadrant II) Sin u = Sin v = Cos u = Cos v = Tan u = Tan v = Sin (u+v) cos (u+v) tan (u+v) Sin (u v) Sin (u v) Sin (u v) Sin (v+u) cos (v+u) tan (v+u) Sin (v u) Sin (v u) Sin (v u)

7 2. Find the exact value of the trigonometric function given that and. (Both u and v are in Quadrant II) Sin u = Sin v = Cos u = Cos v = Tan u = Tan v = Sin (u+v) cos (u+v) tan (u+v) Sin (u v) Sin (u v) Sin (u v) Sin (v+u) cos (v+u) tan (v+u) Sin (v u) Sin (v u) Sin (v u) Additional Problems on extra videos Extra video #3 covers the following problems. Use these as extra practice and the videos as a guide if you need help. 3. Cos(v u) for #1 above 4. Tan(u + v) for #2 above PQ problems:

8 PT problems: Trig of an inverse trig function (use sum and difference to prove) This concept has THREE types of problems, all solved in a very similar way Type 1 Directions: Write the trigonometric expression as an algebraic expression. 1. *arcsin is another notation for sin -1 x Write like sin(u+v) and identify u and v U= V= Take the trig function of each side to get rid of the inverse trig function U=arcsinx turns into V=arccosx turns into Draw a triangle for each piece and label it appropriate with the given information. Use Pythagorean Theorem as needed. It s okay for there to be variables still in the problem!!! Use the sum and difference formulas to simplify

9 Type 2 Directions: Find the value of the expression without using a calculator. 2. Write like sin(u+v) and identify u and v U= V= 3. Use your knowledge of the Unit Circle to identify what u and v actually are. Plug into sum and difference formulas and simplify. 4. Type 3 Directions: Use Right triangles to evaluate the expression. 5. Write like sin(u+v) and identify u and v U= V= Take the trig function of each side to get rid of the inverse trig function U=cos -1 3/5 turns into V= sin -1 5/13 turns into Draw a triangle for each piece and label it appropriate with the given information. Use Pythagorean Theorem as needed. Use the sum and difference formulas to simplify

10 Additional Problems on extra videos Extra video #4 covers the following problems. Use these as extra practice and the videos as a guide if you need help. 6. cos(arccos x arcsin x) Extra video #5 covers the following problems. Use these as extra practice and the videos as a guide if you need help Extra video #6 covers the following problems. Use these as extra practice and the videos as a guide if you need help. 10. PQ problems: Evaluate the expression PT problems: Evaluate the expression Simplifying sum and difference identities Expand the expression using sum and difference identities. Combine like terms and simplify. 1.

11 Additional Problems on extra videos Extra video #7 covers the following problems. Use these as extra practice and the videos as a guide if you need help PQ problems: Simplify or verify the trigonometric identity

12 PT problems: and * Mrs. Kirch couldn t figure this one out when she tried if you can, you get extra credit! Turn in your work page to the box as soon as possible. Extra credit is limited to the first person from each class period to submit! Solving trig equations with sum and difference Expand the expression using sum and difference identities. Combine like terms and simplify as much as you can. Solve the following equation from [0,2π) 1. cos (x + π/4) + cos (x π/4) = 0 2. Additional Problems on extra videos Extra video #7 covers the following problems. Use these as extra practice and the videos as a guide if you need help. 3. PQ problems: Solve the equation from [0,2π) 4. PT problems: Solve the equation from [0,2π) 5.