Right Triangle Trigonometry Section 5.2

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

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

Function Name AbbreviationValue Sinesinopposite/hypotenuse Cosinecosadjacent/ hypotenuse Tangenttanopposite/adjacent Cotangentcotadjacent/opposite Secantsechypotenuse/adjacent Cosecantcschypotenuse/opposite

SOHCAHTOA Some old hippie cut another hippie tripping on apple.

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

Fundamental Identities Sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ cot 2 θ + 1 = csc 2 θ

Ex: The sine of 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 a 2 + b 2 = c b 2 = b 2 = 25 b 2 = 16 b = 4 cos θ = b/c = 4/ 5 tan θ = a/b = 3/ 5 cot θ = b/a = 5/ 3 sec θ = c/b = 5/4 csc = c/a = 5/3

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 a 2 + b 2 = c = c = c 2 c = √10 sin θ = a/c = 1/ √10 = √10/10 cos θ = 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

Discovery time

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 cos 80˚ sin л/3 is equal to cos (л/2 ‒ л/3) cos л/4 is equal to sin (л/2 ‒ л/4) csc л/5 is equal to sec (л/2 ‒ л/5)

Θ(Degrees)Θ(Radians) sin θ = cos(90˚ ‒ θ) sin θ = cos(л/2 ‒ θ) cos θ = sin(90˚ ‒ θ) cos θ = sin(л/2 ‒ θ) tan θ = cot(90˚ ‒ θ) tan θ = cot(л/2 ‒ θ) cot θ = tan(90˚ ‒ θ) cot θ = tan(л/2 ‒ θ) sec θ = csc(90˚ ‒ θ) sec θ = csc(л/2 ‒ θ) csc θ = sec(90˚ ‒ θ) csc θ = sec(л/2 ‒ θ)

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

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

Sec 5.2 HW all all all all