Pre-Calculus Unit 3 Standards-Based Worksheet

Transcription

1 Pre-Calculus Unit 3 Standards-Based Worksheet District of Columbia Public Schools Mathematics STANDARD PCT.P.9. Derive and apply basic trigonometric identities (e.g., sin 2 θ+cos 2 θ= 1,tan 2 θ + 1 = sec 2 θ) and the laws of sines and cosines. Concepts: trigonometric identities (e.g., sin 2 θ+cos 2 θ= 1, tan 2 θ + 1 = sec 2 θ) law of sines law of cosines Skills: derive (basic trigonometric identities, law of sines, law of cosines) apply (basic trigonometric identities, law of sines, law of cosines) Big Ideas: Trigonometric identities can be derived using the Pythagorean Theorem. The laws of sines and cosines can be used to find sides and angles of non-right triangles. The laws of sines and cosines are often used in real-world situations, such as surveying. Essential Questions: How do trigonometric identities lead to other properties of trigonometry? How can you derive tan 2 θ + 1 = sec 2 θ or 1 or cot 2 θ = csc 2 θ from sin 2 θ+cos 2 θ= 1? Under what conditions would you use the law of sines? Under what conditions would you use the law of cosines? Why can you sometimes use the laws of sines or cosines and end up with two different answers? Engaging Scenario: Two neighbors have a due north-south boundary between their land. The neighbor on the east side has cut down a tree that he claims was on his land, and therefore his to cut. The neighbor on the west side thinks the tree was on his land, and has asked you to testify on his behalf. The land title shows that the boundary runs due north-south on a line 251 meters due east of a USGS marker located on the western neighbor s land. You measure the stump to be 270 meters from the marker at a bearing of 76º from North. Is your client correct that the tree was actually on his land? Prepare a written report of your findings that includes mathematical evidence for your conclusion.

2 N Tree somewhere in this circle 76 USGS Marker N-S Property Line PERFORMANCE TASKS Students recognize the triangle formed by the tree, the marker, and the boundary line. THINKING SKILLS ( )* draw (comprehension) PERFORMANCE TASK ASSESSMENT (PROFICIENT CRITERIA) Students draw a triangle to represent the situation, labeling each side and the angles. They also can determine that the angle in the triangle is 14, not 76 degrees. tree 270 USGS marker C Students recognize that the location of the tree depends on the measure of angle C. recognize (analysis) Students recognize that the value of angle C can be used to decide on whose land the tree stood. If C < 90 then the tree stood on the western neighbor s land.

3 Students determine the measure of angle C. determine (application) Students determine the measure of angle C. Although different approaches are possible, one solution would be the following: Students solve for the side opposite the 14 degree angle using the law of cosines (side = ). Students summarize their work in a written report that describes their analysis and explains their reasoning. summarize (evaluation) Students solve for angle C by properly using the Law of Sines (C = 80.5). Students write a convincing summary that includes mathematical support and the conclusion that the tree was actually on the western neighbor s land. *See Bloom s Taxonomy of Thinking Skills to determine higher order thinking skills () or lower order thinking skills (). The goal is to create tasks that employ higher order thinking skills. Standard Assessment: 1) What is the Pythagorean Property for cosine and sine? a) sin 2 θ cos 2 θ = 1 b) sin 2 θ = 1 + cos 2 θ c) cos 2 θ = 1+ sin 2 θ d) sin 2 θ + cos 2 θ = 1 e) cos 2 θ 1 = sin 2 θ (Adapted from Precalculus, page 155, #1) 2) Find angle G in MEG, if m = 5 cm, e = 6 cm, and g = 8 cm. (Adapted from Precalculus, page 230, #6) 3) To find the distance between two points A and B, a surveyor chooses a point C that is 420 yards from A and 540 yards from B. If angle ACB has a measure of 63.1, then approximate the distance between A and B. (Swokowski and Cole Precalculus Functions and Graphs, 7 th Ed, page 484, #12) 4) tanθ + cotθ = tanθ 2 2 a) 1 b) csc θ c) sin θ d) cotθ e) (McGraw Hill s SAT II Math Level II. page 221, #48) 2 sec θ

4 5) A triangle has sides measuring 4, 4, and 6 inches. What is the measure of the largest angle? a) 82.8 b) 97.2 c) 41.4 d) e) (McGraw Hill s SAT II Math Level II. page 264, #16) 6) (6sin x)(3sin x) (9cos x)( 2cos x) = a) 1 b) -18 c) 18 d) -1 e) (McGraw Hill s SAT II Math Level II. page 268, #35) 18sin x 18cos 2 2 x 7) A surveyor needs to determine the length of a pond. He sets up three posts labeled,, and, as shown in the diagram above. The surveyor can measure angles,, and and the lengths of segments and. What is the MINIMUM number of these measurements the surveyor would need to make in order to determine the distance across the pond from post to post? a) 2 b) 3 c) 4 d) 5 (Springboard, Pre-calculus Sequenced Diagnostic Assessment 4: Trigonometric Applications and Identities, #7)

5 Resources: Textbook Materials: Precalculus. Key Curriculum Press, Sections 4-1, 6-2, 6-4. Precalculus Instructor s Resource Book, Key Curriculum Press, Exploration 4-2a (pp ), Exploration 6-2a, 6-2b (pp ), Exploration 6-4a, 6-4b (97, 98). Springboard PreCalculus With Data Analysis, The College Board, Discus Throw. Supplementary Materials: (A good resource for lesson plans.) (A talk that John Mahoney gave on Benjamin Banneker and his work with Trigonometry, provides a good tie in to local history and mathematics.) (Includes trig lessons on the law of sines and cosines.) (Two links to lesson plans using laws of sines and cosines.) (Provides online diagnostic assessments.) Diehl, John J. (Ed). SAT II Math Level 2. McGraw Hill (Good source of multiple choice problems for pre-calc.)

6 Pre-Calculus Unit 3 Standards-Based Worksheet District of Columbia Public Schools Mathematics STANDARD PCT.P.10. Demonstrate an understanding of the formulas for the sine and cosine of the sum or the difference of two angles. Relate the formulas to DeMoivre s theorem and use them to prove other trigonometric identities. Apply to the solution of problems. Concepts: formulas: sine of two angles, difference of two angles DeMoivre s theorem trigonometric identities Skills: demonstrate (understanding of the formulas) use (formulas) prove (other trigonometric identities) Big Ideas: Mathematical relationships often lead to formulas that are useful for solving problems. The sine and cosine of any angle can be found using the sum or difference of two other angles. DeMoivre s Theorem provides a shortcut to finding the sine or cosine of multiples of an angle. Trigonometric ratios for sums, differences, double and half-angles depend on knowing only sine and cosine ratios. Essential Questions: Why use and prove trigonometric identities? How do identities help us to find exact values for trigonometric ratios of angles of than multiplies of 30 and 45? Why doesn t the cos (a + b) = cos a + cos b? How do identities help us to establish other mathematical relationships? Engaging Scenario: A) Movie theater designers would like the angle between the top and the bottom of the movie screen to be less than 35 so moviegoers will be comfortable watching the movie. You work for a movie theater company. A contractor has brought you the following sketch for the new theater. He is concerned that the front row of seats might be too close to the screen. Use a trigonometric difference identity to determine if the contractor is right. The distance from the movie screen is 18 feet. The screen hangs 8 feet off the ground and is 18 feet tall.

7 18 ft 8 ft 18 ft B) You did such great work for the company the owner hires you for the creation of a new theater. This time he wants to put in a bigger screen. If the angle angle between the top and the bottom of the movie screen needs to be less than 35, and the front row will still be 18 feet away, what is the maximum height the screen can be? Bonus: Is their ever a situation when the angle between the top and bottom of the screen will be greater than 35 degrees? Explain your reasoning with sketches and trigonometric identities. PERFORMANCE TASKS Student recognizes two right triangles within the picture and the two angle values they need to find. Students identify the trigonometric difference identity they can use to solve the problem. Students calculate the angle using the formula they found and the proper trigonometric ratios. THINKING SKILLS ( )* recognize (comprehension) identify (comprehension) calculate (application) PERFORMANCE TASK ASSESSMENT (PROFICIENT CRITERIA) Students recognize the two right triangles within the picture, labeling the angles they need to find and the angles that they know. Students identify the need to use the tan of the difference of two angles [tan(b-a)]. Students could also use sin(b-a) or cos(b-a) if they first compute the length of the hypotenuse of each right triangle. Students calculate the difference between the two angles using an appropriate difference identity. Students should get the difference to be approximately 31.3

8 Students prepare a written analysis of their findings. Students modify their triangle to find the maximum height of the movie screen. prepare (analysis) modify (synthesis) Students prepare an analysis including accurate drawings, complete calculations and an analysis explaining their findings. Students modify their triangle, using the information they have, to find the maximum possible height of the screen (they can use 35 as the angle, but need to have x for the height of the movie theater). Students will most likely use triangle trigonometry to solve this problem. For example, the angle between the bottom of the screen and the floor is approximately 24. Students can set up and solve this equation to find screen height: o o tan( ) = x Students assess the situation to determine whether or not it is possible to achieve an angle greater than 35 by altering the distance from the screen. assess (evaluation) Students assess how changing the distance affects the triangles. Students should see that as the angle of the smaller triangle changes, the difference never goes above 35 degrees. In a more advanced evaluation of the problem, students might note that altering the height of the screen can create situations where the angle is greater than 35 degrees. *See Bloom s Taxonomy of Thinking Skills to determine higher order thinking skills () or lower order thinking skills (). The goal is to create tasks that employ higher order thinking skills. Standard Assessment: 1 1) If cos2θ = 3/4, then = 2 2 cos θ sin θ a) -1 b) 3/4 c) 4/3 d) 4 e) 1 (McGraw Hill s SAT II Math Level II. page 241, #16) 2) Express cos13 cos50 sin13 sin50 as a trigonometric function of one angle. (Swokowski and Cole Precalculus Functions and Graphs, 7 th Ed, page 429, #12)

9 3) Suppose that cos A = 0.6, sin A = 0.8, cos B = 0.96 and sin B = Use the composite argument property to find cos (A + B). (Precalculus Instructors Resource Book, page 342, #10) 4) The double-angle formula can be used to determine the value of. What is this value? a) b) c) d) (Springboard, Pre-calculus Sequenced Diagnostic Assessment 4: Trigonometric Applications and Identities, #8) 5) Use the composite argument property to write y = 7cos(θ 20) as a linear combination of cos θ and sin θ. (Precalculus Instructors Resource Book, page 342, #18) Resources: Textbook Materials: Precalculus. Key Curriculum Press, Chapter 5. Precalculus Instructor s Resource Book. Key Curriculum Press, Chapter 5. Springboard Mathematics Precalculus With Data Analysis, The College Board, Seeing Is Believing. Supplementary Materials: (A good resource for lesson plans.) (Demonstrates how to derive the sum and difference angles using the Pythagorean theorem.) (Gives an interesting way to teach trigonometry and laws of sine and cosine using spaghetti.) (A fun story to help students remember the formulas for the sums of sine and cosine.) (A great connection between math and chemistry using trig.)