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Lesson 12 RCSD Geometry Local MATHEMATICS CURRICULUM Name____________________________ U8 Date_____________ Period _________ Lesson 12: Defining Trigonometric Ratios Learning Targets ο· For a given acute angle of a right triangle, I can identify the opposite side, adjacent side, and hypotenuse. ο· Given the side lengths of a right triangle with acute angles, I can find sine, cosine, and tangent of each acute angle Opening Exercise (4 minutes) Use the right triangle βπ΄π΅πΆ to answer 1β3. 1. Name the side of the triangle opposite β π΄. 2. Name the side of the triangle opposite β π΅. 3. Name the side of the triangle opposite β πΆ. New concept: The triangle below is right triangle βπ΄π΅πΆ. Denote β π΅ as the right angle of the triangle. ο· Mark β π΄ in the triangle with a single arc. With respect to acute angle β π΄ of a right triangle β³ π΄π΅πΆ, we identify the opposite side, the adjacent side, and the hypotenuse as follows: ο· Μ Μ Μ Μ , With respect to β π΄, the opposite side, denoted πππ, is side π΅πΆ With respect to β π΄, the adjacent side, denoted πππ, is side Μ Μ Μ Μ π΄π΅ , ο· The hypotenuse, denoted βπ¦π, is side Μ Μ Μ Μ π΄πΆ and is opposite from the 90Λ angle. ο· Example 1 For each exercise, label the appropriate sides as adjacent, opposite, and hypotenuse, with respect to the marked acute angle. a. b. c. Lesson 12 RCSD Geometry Local MATHEMATICS CURRICULUM Name____________________________ Date_____________ Period _________ Example 2 1. Identify the 2. Identify the 3. Identify the πππ πππ ππ π πππ πππ ππ π ratio for angles β π¨ ratio for angles β π¨. ratio for angles β π¨ In any right triangle, the ratios found above are called trigonometric ratios To remember the Basic Trigonometric Functions we use (SOH β CAH β TOA) Example 3. In βπ·πΈπΉ, πβ π· = ππ. π° and πβ πΈ = ππ. π°. Complete the following table. Measure of Angle ππ. π ππ. π πππ πΊπππ π½ = ( ) πππ U8 ππ π πͺπππππ π½ = ( ) πππ π»ππππππ π½ πππ =( ) ππ π Lesson 12 RCSD Geometry Local MATHEMATICS CURRICULUM Name____________________________ U8 Date_____________ Period _________ Now you try Example 4 . In the triangle below, πβ π¨ = ππ. π° and πβ π© = ππ. π°. Complete the following table. Measure of Angle Sine Cosine Tangent ππ. π ππ. π Example 5 . In the triangle below, let π be the measure of β π¬ and π be the measure of β π«. Complete the following table. Measure of Angle Sine Cosine π π Closing (3 minutes) Describe the ratios that we used to calculate sine, cosine, and tangent. a. Given an angle, π, sin π = , cos π = , and tan π = Tangent Name____________________________ U8 Lesson 12 RCSD Geometry Local MATHEMATICS CURRICULUM Date_____________ Period _________ Lesson 12: Defining Trigonometric Ratios Classwork 1. In the triangle below, let π be the measure of β πΏ and π be the measure of β π. Complete the following table. Measure of Angle Sine Cosine Tangent π π 2. Given the diagram of the triangle, complete the following table. Angle Measure π¬π’π§ π½ ππ¨π¬ π½ πππ§ π½ π π a. Which values are equal? b. How are πππ§(π) and πππ§(π) related? 2 3. If π’ and π£ are the measures of complementary angles such that sin π’ = 5 and tan π£ = and angles of the right triangle in the diagram below with possible side lengths. β21 , 2 label the sides Lesson 12 RCSD Geometry Local MATHEMATICS CURRICULUM Name____________________________ U8 Date_____________ Period _________ Lesson 12: Defining Trigonometric Ratios Homework 1. Given the triangle in the diagram, complete the following table. π¬π’π§ Angle Measure ππ¨π¬ πππ§ πΌ π½ 2. Given the table of values below (not in simplest radical form), label the sides and angles in the right triangle Angle Measure πΌ π½ sin 4 cos 2β6 tan 4 2β10 2β6 2β10 4 2β10 2β10 2β6 2β6 4 3. Given sin πΌ and sin π½, complete the missing values in the table. You may draw a diagram to help you. Angle Measure πΌ sin β2 cos 5 3β3 3β3 tan π½ ********4. Given the triangle shown to the right, fill in the missing values in the table. Hint: Use the Pythagorean Theorem to find hypotenuse: Angle Measure πΌ π½ π¬π’π§ ππ¨π¬ πππ§