The Tangent Ratio

1. The Tangent Ratio Chapter 7 Right Triangle Trigonometry

2. You will need a protractor for this activity 1. Each person draw a right triangle (∆ABC) where ﮮ A has a measure of 30º. 2. Each person in the group should draw the triangle with different side lengths, then measure the legs using inches. 3. Compute the ratio leg opposite ﮮ A leg adjacent ﮮ A 4. Compare the ratio with the others in the group. Make a conjecture.

3. Trigonometry and the Tangent Ratio • Objectives: • Use tangent ratios to determine side lengths in triangles • Previously, to find measures in a right triangle, we used: • Pythagorean Theorem • Distance Formula • 30-60-90 or 45-45-90 special right triangles theorems • Now, we will use Trigonometry (triangle measure). • We will investigate 3 of the 6 trigonometric functions: • tangent • sine • cosine

4. Trigonometry and the Tangent Ratio • Tangent Ratio: • In a right triangle, the ratio of the length of the leg opposite ﮮ P to the length of the leg adjacent to ﮮ P In a right triangle, the ratio of the length of the leg opposite ﮮ P to the length of the leg adjacent to ﮮ P . This is called the tangent ratio. Tangent of ﮮ P = opposite leg adjacent leg

5. opposite adjacent Tangent A = Write a Tangent Ratio It is just a formula!!!! Do Now Write the tangent ratios for ﮮT and ﮮ U. tan T = opp = UV = 3 adj TV 4 tan U = opp = TV = 4 adj UV 3

6. what is a tangent ratio? • We use tangent ratios to determine side lengths and angles in right triangles. • In a right triangle, the ratio of the length of the leg opposite to an angleto the length of the leg adjacent to the same angle. • This ratio is always constant

7. The Tangent Ratio A • Tangent of <A: Length of the leg opposite <A Length of the leg adjacent to <A B C Opp Adj Tan =

8. Writing Tangent Ratios • Write the tangent ratios for <K and <J: J tan<K = 7/3 7 tan<J = 3/7 L K 3

9. Find the Tangent Ratios • Find the tangent ratios for <A and <B: A tan <A = 2/1 or 2 1 tan <B = 1/2 C 2 B

10. Find the Tangent Ratios • Find the tangent ratios for <A and <B: A tan <A = 6/3 or 2 3 tan <B = 3/6 or 1/2 C 6 B

11. Solve for the missing side • Find the value of w to the nearest tenth: 10 Start at 54°. We have sides opposite and adjacent of that angle. We can use tangent to solve for the missing side. 54° w Set up the tangent ratio: tan 54 = w 10 Cross multiply w = (tan 54)(10) w = 13.8

12. Solve for the missing side • Find the value of w to the nearest tenth: Start at 57°. We have sides opposite and adjacent of that angle. We will use tangent to solve for the missing side. w Set up the tangent ratio: tan 57 = w 2.5 2.5 57° Cross multiply w = (tan 57)(2.5) w = 3.8

13. Solve for the missing side • Find the value of w to the nearest tenth: 1 Start at 28°. We have sides opposite and adjacent of that angle. We will use tangent to solve for the missing side. w Set up the tangent ratio: tan 28 = 1 w 28° Cross multiply (tan 28)(w) = 1 Divide by tan 28 w = 1 tan 28 w = 1.9

14. The Tangent Ratio Solving for Angle Measures Chapter 7 Right Triangle Trigonometry

15. Using the Inverse of Tangent • The Inverse Tangent Button on your calculator looks like this: tan-1 (You must press the “2nd” or “SHIFT” or “INV” Button and then press “tan”) *We use inverse tangent when solving for a missing angle measure

16. Using Inverse Tangent • Find the m < X to the nearest degree: We will use tangent to find the measure of <X H Set up the tangent ratio: tan X = 5 8 5 Use inverse tangent: X = tan-1 5 8 8 X B X = 32° If no calculator: divide 5/8, =0.6250 Look under the Tangent column of the trig table for the number closest to 0.6250, then move your finger to the left to find the degrees (32) under the Angle column.

17. Using Inverse Tangent • Find the m< Y to the nearest degree: Y We will use tangent to find the measure of <Y Set up the tangent ratio: tan Y = 23 25 25 Use inverse tangent: Y = tan-1 23 25 Y = 43° T P 23