The Primary Trigonometric Ratios Trigonometry: The study of triangles (sides and angles) Trigonometry has been used for centuries in the study of: surveying astronomy geography engineering physics

Parts of a Right Triangle B hypotenuse opposite A C adjacent

B hypotenuse adjacent opposite C A

B hypotenuse opposite A adjacent C

B hypotenuse adjacent C A opposite

B hyp opp A C adj SOH CAH TOA

SOH CAH TOA B hyp 10 opp 8 A C 6 adj

B hyp SOH CAH TOA 5 adj 3 A C 4 opp

SOH CAH TOA B hyp 13 5 adj A 12 C opp

Use a calculator to determine the following ratios. sin 21° = 0.3584 cos 53° = 0.6018 tan 72° = 3.0777

sin A = 0.4142 ÐA = sin-1(0.4142) = 24° cos B = 0.6820 Determine the following angles (nearest degree). sin A = 0.4142 ÐA = sin-1(0.4142) = 24° cos B = 0.6820 ÐB = cos-1(0.6820) = 47° tan C = 1.562 ÐC = tan-1(1.562) = 57°

Determine the following angles (nearest degree). sin A = ÐA = sin-1(0.5833) = 36° = 0.5833 cos B = ÐB = cos-1(0.2666) = 75° = 0.2666 ÐC = tan-1(1.875) tan C = = 62° = 1.875

a = 6 sin 30° a = 6 (0.5) a = 3 cm B Example 1: Determine side a opp hyp SOH CAH TOA 6 cm a 30º A C a = 6 sin 30° a = 6 (0.5) a = 3 cm

Ex. 2: Name two trig ratios that will allow us to calculate side b. 50º 9 m 40º A b C

Example 3: Determine side b SOH CAH TOA 55º 8 cm adj A b C opp b = 8 tan 55° b = 8 (1.428) b = 11.4 cm

Example 4: Determine the measure of ÐP. SOH CAH TOA R adjacent cos P = 0.70588 12 cm ÐP = cos–1(0.70588) Q 17 cm ÐP = 45.1° P hypotenuse

Example 5: Determine the measure of side PR. Method 1 adj opp R q 12 cm q(tan 35°) = 12 35° P Q q = 17.1 cm

q = 17.1 cm Example 6: Determine the measure of side PR. Method 2 adj opp R ÐQ = 90° – 35° q ÐQ = 55° 12 cm 35° 55° P Q q = 12(tan 55°) q = 12(1.428) q = 17.1 cm

Ex. 7: In DPQR, ÐQ = 90°. 8 cm b) Find cos R . 4 cm R a) Find sin R if PR = 8 cm and PQ = 4 cm. R P Q 8 cm b) Find cos R . 4 cm RQ2 = 82 – 42 RQ2 = 64 – 16 RQ2 = 48