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Bell Ringer

Bell Ringer. Tangent Ratios. A trigonometric ratios is a ratio of the lengths of two sides of a right triangle. For any acute angle of a right triangle, there is a leg opposite the angle and a leg adjacent to the angle. The ratio of these legs is the tangent of the angle. Example 1.

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Bell Ringer

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  1. Bell Ringer

  2. Tangent Ratios • A trigonometric ratios is a ratio of the lengths of two sides of a right triangle. • For any acute angle of a right triangle, there is a leg opposite the angle and a leg adjacent to the angle. The ratio of these legs is the tangent of the angle.

  3. Example 1 SOLUTION leg opposite S 4 tan S = = = ≈ 1.7321 4 leg adjacent to S leg opposite R 1 4 3 3 3 3 = = tan R = ≈ 0.5774 leg adjacent to R 4 Find Tangent Ratio Find tan S and tan R as fractions in simplified form and as decimals rounded to four decimal places.

  4. Example 2 SOLUTION Rounded value Calculator keystrokes Display 3.487414444 3.4874 Use a Calculator for Tangent Approximate tan74° to four decimal places. or 74 74

  5. Now you Try  Find Tangent Ratio Find tan Sand tan R as fractions in simplified form and as decimals. Round to four decimal places if necessary. 1. tan S = tan R = ≈1.3333 =0.75; ANSWER 12 3 4 5 =2.4 tan S = tan R = ≈0.4167; 12 3 4 5 ANSWER 2.

  6. Checkpoint Now you Try  Find Tangent Ratio Use a calculator to approximate the value to four decimal places. 3. tan 35° 4. tan 85° 11.4301 0.7002 0.1763 ANSWER ANSWER ANSWER 5. tan 10°

  7. Example 3 Find Leg Length Use a tangent ratio to find the value of x. Round your answer to the nearest tenth. SOLUTION tan 22° = Write the tangent ratio. Substitute. 3 tan 22° = x opposite leg x· tan 22° = 3 Multiply each side by x. adjacent leg Divide each side by tan22°. x = 3 Use a calculator or table to approximate tan22°. tan 22° x ≈ 3 x≈ 7.4 Simplify. 0.4040

  8. Example 4 Method 1 Method 2 Find Leg Length Use two different tangent ratios to find the value of x to the nearest tenth. SOLUTION First, find the measure of the other acute angle: 90° – 35° = 55°. tan 55° = tan 35° = x 4 tan 55° = tan 35° = x 4 opposite leg opposite leg adjacent leg adjacent leg x· tan 35° = 4 4 tan 55° = x

  9. Example 4 Find Leg Length 4(1.4281)≈x x≈ 5.7 x≈ 5.7 The two methods yield the same answer: x≈5.7. ANSWER x = 4 tan 35° x ≈ 4 0.7002

  10. Example 5 Estimate Height You stand 45 feet from the base of a tree and look up at the top of the tree as shown in the diagram. Use a tangent ratio to estimate the height of the tree to the nearest foot. SOLUTION tan 59° = Write ratio. h tan 59° = 45 opposite leg Substitute. adjacent leg 45 tan 59° = h Multiply each side by 45. 45(1.6643)≈h Use a calculator or table to approximate tan 59°. 74.9≈h Simplify.

  11. Example 5 Estimate Height The tree is about 75 feet tall. ANSWER

  12. Checkpoint ANSWER 8 tan 44° = and tan 46° = x Now you Try  Find Side Length Write two equations you can use to find the value of x. 6. ANSWER x 4 5 x x 7. x x 8 4 5 tan 37° = and tan 53° = ANSWER 8. tan 59° = and tan 31° =

  13. Checkpoint Now you Try  Find Side Length Find the value of x. Round your answer to the nearest tenth. 9. 10. 12.6 10.4 34.6 ANSWER ANSWER ANSWER 11.

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  15. Page 560 #s 2-30

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