Right Triangle Trigonometry

1. Right Triangle Trigonometry Section 5.2

2. Right Triangle Recall that a triangle with a 90˚ is a right triangle

3. There are six ratios between the hypotenuse and two legs of a right triangle. Sine, cosine, tangent, cotangent, secant, and cosecant.

5. SOHCAHTOA Some old hippie cut another hippie tripping on apple.

6. Reciprocal Identities cscθ = 1/sinθ sec θ = 1/cosθ cotθ = 1/tan tanθ = sinθ/cosθ cotθ = cosθ/sinθ

7. Fundamental Identities Sin2 θ + cos2θ = 1 tan2 θ + 1 = sec2 θ cot2 θ + 1 = csc2 θ

8. Ex: The sineof an acute angle of a right triangle is 3/5. Find the exact value of each of the remaining five trigonometric functions. sin θ = opp/hyp = a/c = 3/5 so a=3 and c=5 a2 + b2 = c2 32+ b2= 52 9 + b2 = 25 b2 = 16 b = 4 cosθ = b/c = 4/5 tanθ = a/b = 3/5 cot θ = b/a = 5/3 sec θ = c/b = 5/4 csc = c/a = 5/3

9. Ex: The tangent of an acute angle of a right triangle is 1/3. Find the exact value of each of the remaining five trigonometric functions. tan θ = opp/adj = a/b = 1/3 so a=1 and b=3 a2 + b2 = c2 12+ 32= c2 1 + 9 = c2 c = √10 sin θ = a/c = 1/√10= √10/10 cosθ = b/c = 3/√10 = (3√10)/10 cot θ = b/a = 3/1= 3 sec θ = c/b = √10/3 csc = c/a = √10/1 = √10

10. Discovery time

11. Complementary angle theorem Cofunctions of complementary angles are equal. For example sin 30˚ is equal to cos 60˚ sin 20˚ is equal to cos 70˚ sin 10˚ is equal to cos80˚ sin л/3 is equal to cos(л/2 ‒ л/3) cosл/4 is equal to sin (л/2 ‒ л/4) cscл/5 is equal to sec (л/2 ‒ л/5)

13. Using the Complementary Angle Theorem Example 7b (page 399): Find the exact value of = = = 1

14. Another example: Find the exact value of = = = 1

15. Sec 5.2 HW 11-16 all 25-26 all 37-42 all 55-60 all