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Mr. Wolf Monday 12/8/08 Geometry Grades 10-12 Unit 7: Right Triangles Inverse Trig Functions & Real World Applications Materials and Resources: Warm-up (1 per student) Inverse Trigonometric Functions sheet (1 per student) Exit Ticket (1 per student) PA Standards Addressed: Instructional Objectives: Students will be able to evaluate inverse trigonometric functions. Students will be able to solve for missing angles in right triangles. Students will be able to solve real world applications of right triangle trigonometry. Time 10 min 1 min 40 min Activity Warm-up Agenda Inverse Trig Functions Pg. 308 #11-12 Pg. 314 #1, 2, 6 35 min Real World Applications Pg. 319 #4, 5 1 min 5 min Agenda Conclusion Homework: Pg. 308 #5-9 Pg. 314 #3-5, 7, 8 Pg. 319 #3, 6 Lesson Reflection: Description Pass out the Warm-up and review solutions. Review the goals for the day. Modeling: Guiding: Independent Practice: Assessment: Modifications: Students with special needs… Advanced students… Modeling: Guiding: Independent Practice: Assessment: Modifications: Students with special needs… Advanced students… Revisit goals and identify whether they were met. Pass out the Exit Ticket and collect at the bell. Geometry Fall 2008 Name: ________________________ Warm-up Use a calculator to find the value of the following trigonometric functions. Round answers to the nearest hundredth. sin(24°) = cos(55°) = tan(80°) = Use a calculator to find the value of x. Round answers to the nearest hundredth. cos(35°) = x 7 x = ________ tan(12°) = 4 x sin(68°) = x = ________ x 8 x = ________ Find the values of x in the following triangles. Geometry Fall 2008 Name: ________________________ Warm-up Use a calculator to find the value of the following trigonometric functions. Round answers to the nearest hundredth. sin(24°) = cos(55°) = tan(80°) = Use a calculator to find the value of x. Round answers to the nearest hundredth. cos(35°) = x 7 x = ________ tan(12°) = 4 x x = ________ Find the values of x in the following triangles. sin(68°) = x 8 x = ________ Geometry Fall 2008 Name: ________________________ Inverse Trigonometric Functions Recall that sin(x), cos(x), and tan(x) are all functions of x. Therefore, we want to examine the inverse functions. Trigonometric Functions Basic Operations Function Inverse Function Operation Inverse Operation Sine Inverse Sine Addition Subtraction sin(x) sin-1(x) + – Cosine Inverse Cosine Multiplication Division cos(x) cos-1(x) • ÷ Tangent Inverse Tangent tan(x) tan-1(x) To evaluate the inverse sine, cosine, or tangent with a graphing calculator: 1) Hit the key in order to activate the secondary commands 2) Hit the , , or wish to evaluate 3) Enter the angle measure 4) Hit key depending on which inverse function you Examples: Evaluate the following trigonometric inverse functions. Round answers to the nearest whole number. sin-1(0.4068) = cos-1(0.5736) = tan-1(5.6713) = Examples: Solve the following equations for x. Round answers to the nearest whole number. sin(x) = 0.9063 x = ______ cos(x) = 0.9397 tan(x) = 1 x = ______ x = ______ Finding Unknown Angle Measures To find an unknown angle measure… 1) Set up a sine, cosine, or tangent proportion 2) Use the inverse sin, cos, tan button on your calculator and solve for the missing value. Example: Solve for x. Geometry Fall 2008 Name: ________________________ Real World Applications Suppose an operator at the top of a lighthouse sights a sailboat on a line that makes a 25º angle with a horizontal line. The angle between the horizontal and the line of sight is called an angle of depression. At the same time, a person in the boat must look 25º above the horizontal to see the tip of the lighthouse. This is called an angle of elevation. If the top of the lighthouse is 40m above sea level, the distance x between the boat and the base of the lighthouse can be calculated. Example: A sled travels from point A at the top of the hill to point B at the bottom. The angle of depression of the hill (or the steepness) is 35º. If the elevation of point A is 75 yards, find the distance the sled travels. (Hint: Draw a diagram first!) Geometry Fall 2008 Name: ________________________ Exit Ticket Bjorn Snurgensen, the most famous Swedish Olympic athlete, one a gold medal at the 1964 Olympics by throwing a javelin 84 meters. At the halfway point of its journey, the javelin reached its maximum height of 25 meters. Calculate the angle of elevation that Bjorn threw the javelin. 1) Draw a figure to represent the above scenario. 2) Label the distance traveled, the halfway point, and the maximum height of the javelin 3) Label the angle of elevation as x 4) Set up a trigonometric ratio and solve for x Geometry Fall 2008 Name: ________________________ Exit Ticket Bjorn Snurgensen, the most famous Swedish Olympic athlete, one a gold medal at the 1964 Olympics by throwing a javelin 84 meters. At the halfway point of its journey, the javelin reached its maximum height of 25 meters. Calculate the angle of elevation that Bjorn threw the javelin. 1) Draw a figure to represent the above scenario. 2) Label the distance traveled, the halfway point, and the maximum height of the javelin 3) Label the angle of elevation as x 4) Set up a trigonometric ratio and solve for x