Lesson 8-5 The Tangent Ratio (page 305) How can trigonometric ratios be used to find sides and angles of a triangle?

Trigonometry, comes from 2 Greek words, which mean “ triangle measurement. ” Our study of trigonometry will be limited to Right Triangle Trigonometry.

The tangent ratio is the ratio of the lengths of the legs. a A b B C c hypotenuse leg

opposite leg vs adjacent leg a A b B C c adjacent leg opposite leg In relationship to angle A …

opposite leg vs adjacent leg a A b B C c opposite leg adjacent leg In relationship to angle B …

Definition of Tangent Ratio a A b B C c tangent of ∠ A = tan A

a A b B C c tangent of ∠ B = tan B

a A b B C c remember

Example 1: Express tan A and tan B as ratios. 17 ____ 15 A B C (a)tan A = ______ (b)tan B = ______ What can we do now? X OH YEAH! I know what I can do! NOT

NEVER, and I mean NEVER, forget my theorem!

Example 1: Express tan A and tan B as ratios. 17 ____ 15 A B C (a)tan A = ______ (b)tan B = ______ Now we can find the ratios! Remember TOA. 8 reciprocals

Example 2 The table on page 311 gives approximate decimal values of the tangent ratio for some angles. (a)tan 20º ≈ ____________ (b)tan 87º ≈ ____________ “≈” means “is approximately equal to” Now try this with a calculator!

To enter this in your calculator you will need to use the TAN function key. (a)tan 20º ≈ ____________ (b)tan 87º ≈ ____________ Enter TAN( 20 ) then press ENTER (=) and round to 4 decimal places. Enter TAN( 87 ) then press ENTER (=) and round to 4 decimal places

Example 3 The table on page 311 can also be used to find an approximate angle measure given a tangent value. (a)tan _______ ≈ (b)tan _______ ≈ “≈” means “is approximately equal to” Now try this with a calculator! 30º 76º

To enter this in your calculator you will need to use the inverse key or 2nd function key. (a)tan _______ ≈ (b)tan _______ ≈ Enter TAN -1 (.5774) then press ENTER (=) and round to the nearest degree Enter TAN -1 (4.0108) then press ENTER (=) and round to the nearest degree 30º 76º

Example 4 (a) Find the value of x to the nearest tenth. x ≈ ________ x 25 37º 18.8 You can type this in your calculator!

Example 4 (b) Find the value of x to the nearest tenth. x ≈ ________ x 3 72º 9.2

Example 4 (c) Find the value of y to the nearest degree. y ≈ ________ 5 4 yº 51º Type this in your calculator!

Example 4 (d) Find the value of y to the nearest degree. y ≈ ________ x 5 yº 32º 8

OPTIONAL Assignment Written Exercises on pages 308 & to 21 odd numbers ~ #22 is BONUS! ~ PK Hint: For #19 on page 309 you should first read Example 3 on page 306. How can trigonometric ratios be used to find sides and angles of a triangle?

The grade of a road is 8%. What angle does the road make with the horizontal? horizontal vertical ROAD angle

The grade of a road is 8%. What angle does the road make with the horizontal? horizontal run = 100 vertical rise = 8 ROAD angle

The grade of a road is 8%. What angle does the road make with the horizontal? horizontal run = 100 vertical rise = 8 ROAD angle The road makes a 5º angle with the horizontal.

Assignment: TRIGONOMETRY WORKSHEET #1 This will be given after the next lesson. Put #19 from page 309 on back of worksheet! How can trigonometric ratios be used to find sides and angles of a triangle?