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7.4 Trigonometry

7.4 Trigonometry. Objectives. Find trigonometric ratios using right triangles Solve problems using trigonometric ratios. Trigonometric Ratios.

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7.4 Trigonometry

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  1. 7.4 Trigonometry

  2. Objectives • Find trigonometric ratios using right triangles • Solve problems using trigonometric ratios

  3. Trigonometric Ratios • The word trigonometry originates from two Greek terms, trigon, which means triangle, and metron, which means measure. Thus, the study of trigonometry is the study of triangle measurements. • A ratio of the lengths of the sides of a right triangle is called a trigonometric ratio. The three most common trigonometric ratios are sine, cosine, and tangent.

  4. Trigonometric Ratios A For right ∆ABC… • sin A = opposite side = a hypotenuse c • cos A = adjacent side = b hypotenuse c • tan A = opposite side = aadjacent side b c b a C B

  5. Trigonometric Ratios A To help you remember these trigonometric relationships, you can use the mnemonic device, SOH-CAH-TOA, where the first letter of each word of the trigonometric ratios is represented in the correct order. Sin A = Opposite sideSOH Hypotenuse Cos A = Adjacent sideCAHHypotenuse Tan A = Opposite sideTOA Adjacent side c b a C B

  6. Example 1: Find sin L, cos L, tan L, sin N, cos N, and tan N. Express each ratio as a fraction and as a decimal.

  7. Example 1:

  8. Example 1:

  9. Example 1:

  10. Answer: Example 1:

  11. Your Turn: Find sin A, cos A, tan A, sin B, cos B, and tan B. Express each ratio as a fraction and as a decimal.

  12. Answer: Your Turn:

  13. Use a calculator to find tan to the nearest ten thousandth. TAN ENTER KEYSTROKES: 56 1.482560969 Answer: Example 2a:

  14. Use a calculator to find cos to the nearest ten thousandth. COS ENTER KEYSTROKES: 90 0 Answer: Example 2b:

  15. Answer: Answer: Your Turn: a. Use a calculator to find sin 48° to the nearest ten thousandth. b. Use a calculator to find cos 85° to the nearest ten thousandth.

  16. Angles of Right Triangles • You can use a calculator or a trigonometric table to find the missing measures of a right triangle if you are given the measures of two sides of the triangleorone side and one acute angle.

  17. EXERCISING A fitness trainer sets the incline on a treadmill to The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor? Example 3: Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches.

  18. SIN ENTER KEYSTROKES: 607 7.312160604 Example 3: Multiply each side by 60. Use a calculator to find y. Answer: The treadmill is about 7.3 inches high.

  19. CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about how high does the ramp rise off the ground to the nearest inch? Your Turn: Answer: about 15 in.

  20. Example 4: COORDINATE GEOMETRY Find mX in right XYZ for X(–2, 8), Y(–6, 4), and Z(–3, 1).

  21. ExploreYou know the coordinates of the vertices of a right triangle and that is the right angle. You need to find the measures of one of the angles. Plan Use the Distance Formula to find the measure of each side. Then use one of the trigonometric ratios to write an equation. Use the inverse to find Solve or Example 4:

  22. or or Example 4:

  23. Example 4: Use the cosine ratio. Simplify. Solve for x.

  24. Use a calculator to find 2ND ) KEYSTROKES: 4 5 ENTER Example 4: Examine Use the sine ratio to check the answer. Simplify.

  25. 2ND ) KEYSTROKES: 3 5 ENTER Answer: The measure of is about 36.9. Example 4:

  26. COORDINATE GEOMETRY Your Turn: Answer: about 56.3

  27. Assignment • Pre-AP Geometry: Pg. 368 #18 – 48, 50, 51, 56 – 58 • Geometry: Pg. 368 #18 – 48

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