Trig right triangle trig

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  • 1.13-1 Right-Angle Trigonometry Holt Algebra 2 Warm Up Lesson Presentation Lesson Quiz

2. Warm Up Given the measure of one of the acute angles in a right triangle, find the measure of the other acute angle. 1.45 2.60 3.24 4.38 45 30 66 52 3. Warm Up Continued Find the unknown length for each right triangle with legsaandband hypotenusec . 5.b= 12,c=13 6.a= 3,b= 3 a = 5 4. Understand and use trigonometric relationships of acute angles in triangles. Determine side lengths of right triangles by using trigonometric functions. Objectives 5. trigonometric function sine cosine tangent cosecants secant cotangent Vocabulary 6. Atrigonometric functionis a function whose rule is given by a trigonometric ratio. Atrigonometric ratiocompares the lengths of two sides of a right triangle. The Greek letter thetais traditionally used to represent the measure of an acute angle in a right triangle. The values of trigonometric ratios depend upon . 7. The triangle shown at right is similar to the one in the table because their correspondingangles are congruent. No matter which triangle is used, the value of sinis the same. The values of the sine and other trigonometric functions depend only on angleand not on the size of the triangle. 8. Example 1: Finding Trigonometric Ratios Find the value of the sine, cosine, and tangent functions for . sin= cos= tan= 9. Check It Out!Example 1Find the value of the sine, cosine, and tangent functions for . sin= cos= tan= 10. You will frequently need to determine the value of trigonometric ratios for 30,60, and 45 angles as you solve trigonometry problems. Recall from geometry that in a 30-60-90 triangle, theration of the side lengths is 1:3 :2, and that in a 45-45-90 triangle, the ratio of the side lengths is 1:1:2. 11. Example 2: Finding Side Lengths of Special Right Triangles Use a trigonometric function to find the value ofx . x= 37 The sine function relates the opposite leg and the hypotenuse. Multiply both sides by 74 to solve for x. Substitutefor sin 30. Substitute 30for , x foropp, and 74 for hyp. 12. Check It Out!Example 2Use a trigonometric function to find the value ofx . The sine function relates the opposite leg and the hypotenuse. Multiply both sides by 20 to solve for x. Substitute 45for , x foropp, and 20 for hyp. Substitutefor sin 45. 13. Example 3: Sports Application In a waterskiing competition,a jump ramp has the measurements shown. Tothe nearest foot, what isthe heighthabove waterthat a skier leaves the ramp? 5 h The height above the water is about 5 ft. Substitute 15.1 for , h for opp., and 19 for hyp. Multiply both sides by 19. Use a calculator to simplify. 14. Make sure that your graphing calculator is set to interpret angle values as degrees. Press. Check thatDegreeand notRadianis highlighted in the third row.Caution! 15. Check It Out!Example 3A skateboard ramp willhave a height of 12 in.,and the angle betweenthe ramp and the groundwill be 17. To the nearest inch, what will be the length l of the ramp? l 41 The length of the ramp is about 41 in. Substitute 17 for ,l for hyp., and 12 for opp. Multiply both sides byland divide by sin 17. Use a calculator to simplify. 16. When an object is above or below another object, you can find distances indirectly by using theangle of elevationor theangle of depressionbetween the objects. 17. Example 4: Geology Application A biologist whose eye level is 6 ft above the ground measures the angle of elevation to the top of a tree to be 38.7. If the biologist is standing 180 ft from the trees base, what is the height of the tree to the nearest foot? Step 1Draw and label a diagram to represent the information given in the problem. 18. Example 4 Continued Step 2Letxrepresent the height of the treecompared with the biologists eye level. Determine the value ofx . Use the tangent function. 180 (tan 38.7) =x Substitute 38.7 for , x for opp., and 180 for adj. Multiply both sides by 180. 144 x Use a calculator to solve for x. 19. Example 4 Continued Step 3Determine the overall height of the tree. x+ 6 = 144 + 6 = 150 The height of the tree is about 150 ft. 20. Check It Out!Example 4A surveyor whose eye level is 6 ft above the ground measures the angle of elevation to the top of the highest hill on a roller coaster to be 60.7. If the surveyor is standing 120 ft from the hills base, what is the height of the hill to the nearest foot? Step 1Draw and label a diagram to represent the information given in the problem. 120 ft 60.7 21. Check It Out!Example 4 Continued Use the tangent function. 120 (tan 60.7) =x Substitute 60.7 for , x for opp., and 120 for adj. Multiply both sides by 120. Step 2Letxrepresent the height of the hillcompared with the surveyors eye level. Determine the value ofx . 214 x Use a calculator to solve for x. 22. Check It Out!Example 4 ContinuedStep 3Determine the overall height of the roller coaster hill. x+ 6 = 214 + 6 = 220 The height of the hill is about 220 ft. 23. The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions,cosecant, secant,andcotangent. 24. Example 5: Finding All Trigonometric Functions Find the values of the six trigonometric functions for . Step 1Find the length of the hypotenuse. a 2+b 2=c 2 c 2= 24 2+ 70 2 c 2= 5476 c = 74 Pythagorean Theorem. Substitute 24 for a and 70 for b. Simplify. Solve for c. Eliminate the negative solution. 70 24 25. Example 5 Continued Step 2Find the function values. 26. In each reciprocal pair of trigonometric functions, there is exactly one co Helpful Hint 27. Find the values of the six trigonometric functions for . Step 1Find the length of the hypotenuse. a 2+b 2=c 2 c 2= 18 2+ 80 2 c 2= 6724 c = 82 Pythagorean Theorem. Substitute 18 for a and 80 for b. Simplify. Solve for c. Eliminate the negative solution. Check It Out!Example 580 18 28. Check It Out!Example 5 ContinuedStep 2Find the function values. 29. Lesson Quiz: Part ISolve each equation. Check your answer. 1.Find the values of the six trigonometric functions for . 30. Lesson Quiz: Part II2.Use a trigonometric function to find the valueofx . 3.A helicopters altitude is 4500 ft, and a planes altitude is 12,000 ft. If the angle of depression from the plane to the helicopter is 27.6, what is the distance between the two, to the nearest hundred feet? 16,200 ft

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