2. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Aim: What does SOHCAHTOA have to do with our study of right triangles? Do Now: Key terms: adjacent, opposite & hypotenuse 4 3 5 CB A What are the following ratios? AC AB = AC CB = BC AB = 3 5 4 5 4 3

3. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Trigonometry Basics - Sine In a right  ABC with right angle  BCA The sine of angle B, written sine B, is defined as 4 3 5 CB A sin B = AC AB = 5 4

4. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Sine’s Reciprocal What is the reciprocal of sin  ? What is the reciprocal of 3?1/3 1/sin  the reciprocal of sin  has a special name: sin  = csc  = ? = ex. using the calculator to find csc 53º: find csc 53º: sin x -1 53 ENTER Display: 1.252135658 Method 1 cosecant

5. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Trigonometry Basics - Cosecant In a right  ABC with right angle  BCA The cosecant of angle B, written csc B, is defined as 4 3 5 CB A csc B = AB AC = 4 5

6. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Trigonometry Basics - Cosine In a right  ABC with right angle  BCA The cosine of angle B, written cos B, is defined as 4 3 5 CB A cos B = BC AB = 5 3 Recall: the sine of an acute angle has the same value as the cosine of its complement.

7. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Cosine’s Reciprocal The reciprocal of cosine  is the secant  : sec  = ? ex. cos  = 2  = ? 2nd ÷ cos ENTER 12 Display: 60 using the calculator to find sec  : Find sec (-38º): ( – ) 1 cos ENTER Display: 1.269018215 Method 2 ÷ 38

8. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Trigonometry Basics - Secant In a right  ABC with right angle  BCA The secant of angle B, written sec B, is defined as 4 3 5 CB A sec B = AB BC = 3 5

9. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Trigonometry Basics - Tangent The tangent of angle B, written tan B, is defined as 4 3 5 CB A In a right  ABC with right angle  BCA tan B = AC BC = 3 4

10. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Tangent’s Reciprocal The reciprocal of tan  is the cotangent  : cot  = ? ex. tan  = =  = ? 2ndtan ENTER Display: 60 Using the calculator to find cot  : Find cot 257º: tanx -1 257 ENTER Display:.2308681911 Method 1 1 tan ENTER Display:.2308681911 ÷ 257 Method 2

11. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Trigonometry Basics - Cotangent The cotangent of angle B, written cot B, is defined as 4 3 5 CB A In a right  ABC with right angle  BCA cot B = BC AC = 4 3

12. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Meet Chief Sine - SOH = Opposite Hypotenuse Tangent - TOA = Opposite Adjacent Cosine - CAH = Adjacent Hypotenuse SOH CAHTOA

13. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Trig. Relationships Recall: the sine of an acute angle has the same value as the cosine of its complement. 4 3 5 C B A sin A = cos B and cos A = sin B The tangent of an acute angle is the reciprocal of the tangent of its complement tan A · tan B = 1 the tangent of an acute angle has the same value as the cotangent of its complement. tan A = cot B and cot A = tan B

14. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Model Problem In right triangle ABC with right angle at C, BC = 6, and AC = 8. Find the three trigonometric functions of  B. 8 6 BC A 10 sin B = cos B = tan B =

15. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Model Problem Park planners would like to build a bridge across a creek. Surveyors have determined that from 5 ft. above the ground the angle of elevation to the top of an 8ft. pole on the opposite side of the creek is 5 o. Find the length of the bridge to the nearest foot. 5’8’ 5o5o x 3’

16. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Model Problems 1.sin 24 o is equivalent to a) cos 24 o b) sin 66 o c) cos 66 0 d) 1/sin 24 0 2. If cot x = tan(x + 20 o ), find x. When the cotangent and tangent functions are equal in value, the angles must be complementary. The sine of an angle has the same value as the cosine of its complement. x + (x + 20) = 90 2x + 20 = 90 x = 70

17. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Degrees, Minutes & Seconds 360 0 in a circle 60 minutes in 1 degree 60 seconds in 1 minute 17 o 43’05” 17 degrees 43 minutes 5 seconds 1 minute is 1/60 th of a degree 1 second is 1/60 th of a minute

18. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Model Problem Find cos 17 o 43’ to 4 decimal places Find sin 20.30 o to 4 decimal places Find sin 20 o 30’ to 4 decimal places

19. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Find An Angle Given a Trig Function Value What is measure of  ? 2ndcos ENTER 32 2nd÷ Calculator’s MODE must be in degrees 30 o What is measure of  ?

20. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Regents Prep In triangle ABC, side a = 7, b = 6, and c = 8. Find m  B to the nearest degree. 1) 43 o 2) 47 o 3) 65 o 4) 137 o

21. Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig. Regents Prep In the diagram below of right triangle KTW, KW = 6, KT = 5, and m  KTW = 90. What is the measure of  K, to the nearest minute? 1) 33 o 33’2) 33 o 55’ 3) 33 o 34’4) 33 o 56’