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Understanding Inverse Functions in Trigonometry: Solving Right Triangles

Explore the application of inverse trigonometric functions to solve for angles in right triangles. Using a practical scenario involving a man on a cliff, we will investigate how to find the distance to a friend below and the angle of elevation. This lesson covers key functions such as sin-1, cos-1, and tan-1, guiding you through examples and practices that enhance your understanding. Learn how to apply these concepts using the unit circle and tackle real-world problems involving angles and shadows, equipping you with invaluable skills in trigonometry.

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Understanding Inverse Functions in Trigonometry: Solving Right Triangles

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  1. Warm up • A man on a 135-ft vertical cliff looks down at an angle of 16 degrees and sees his friend. How far away is the man from his friend? How far is the friend from the base of the cliff?

  2. Lesson 5-5 Solving Right Triangles Objective: To introduce inverse functions and use them to find the angles of right triangles.

  3. Inverse Functions • Inverse functions sin-1 or arcsin • cos-1 or arccos • tan-1 or arctan • They are used to find the missing angle in a right triangle. • THESE ARE NOT THE SAME AS THE RECIPROCAL OF THE FUNCTION

  4. Sin-1 x = • Sin x = .5 is the same as x = arcsin .5 • Use this to solve for the angle

  5. Finding the angle • To find the angle β– use one of the trig functions that you have the info for. Ex: sin β = because we don’t know the angle. β 5 3 α Press [2nd] [sin-1] (4/5) This will give you the degrees of angle β A 4 C

  6. Examples 30o 150o -30o 45o

  7. Find the angle – round to the nearest degree: • Sin A = .9063 • Cos B = .6428 • Tan C = .4040

  8. Using the unit circle • Inverse functions can be evaluated using the unit circle. • Ex: sin-1 ( ) could be either 60o or 120 o.

  9. Find the angles using the unit Circle(Give the angle in quadrant I) 60o 45o 45o

  10. Example Find 28 21

  11. Combining Functions with Inverses • means that you want to find the tangent of whatever angle has a cosine of

  12. Practice • Cos(tan-1(.95)) = • Sin(cos-1(5/6))= • Tan(tan-1(2) = • .725 • .5528 • 2

  13. Practice • Many cities place restrictions on the height and placement of skyscrapers in order to protect residents from completely shaded streets. If a 100-foot building casts an 88-foot shadow what is the angle of the elevation of the sun?

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