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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,

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Presentation on theme: "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,"— Presentation transcript:

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

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

5 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 ANSWER tan S = 3 4 = 0.75 ; tan R = 4 3 ≈ ANSWER tan S = 5 12 ≈ ; tan R = 12 5 = 2.4

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

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° = 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 ≈ Use a calculator or table to approximate tan 22°. x ≈ 7.4 Simplify.

8 Example 4 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°. 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° = 44 tan 55° = x

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

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° = opposite leg adjacent leg Write ratio. tan 59° = h 45 Substitute. 45 tan 59° = h Multiply each side by (1.6643) ≈ h Use a calculator or table to approximate tan 59° ≈ h Simplify.

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

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

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

14 Page 560

15 Page 560 #s 2-30


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