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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.

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

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**Approximate tan 74° to four decimal places.**

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

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

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**Checkpoint Now you Try **

Find Tangent Ratio Use a calculator to approximate the value to four decimal places. 3. tan 35° ANSWER 0.7002 4. tan 85° ANSWER 5. tan 10° ANSWER 0.1763

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

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**First, find the measure of the other acute angle: 90° – 35° = 55°.**

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

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**The two methods yield the same answer: x ≈ 5.7.**

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

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Example 5 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. Estimate Height SOLUTION tan 59° = opposite leg adjacent leg Write ratio. tan 59° = h 45 Substitute. 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. 10

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**The tree is about 75 feet tall.**

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

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**Checkpoint Now you Try **

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

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**Checkpoint Now you Try **

Find Side Length Find the value of x. Round your answer to the nearest tenth. 9. ANSWER 10.4 10. ANSWER 12.6 11. ANSWER 34.6

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A trigonometric ratio is a ratio of the lengths of 2 sides of a right triangle. You will learn to use trigonometric ratios of a right triangle to determine.

A trigonometric ratio is a ratio of the lengths of 2 sides of a right triangle. You will learn to use trigonometric ratios of a right triangle to determine.

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