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:
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 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 = = ≈
Example 2 Use a Calculator for Tangent Approximate tan 74° to four decimal places. SOLUTION Calculator keystrokes 74 or 74 Display Rounded value
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
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
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.
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
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.
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.
Example 5 Estimate Height ANSWER The tree is about 75 feet tall.
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
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