Right Triangle Trigonometry

Section 5.2

Right TriangleRecall 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

Abbreviation Value

Sine sin opposite/hypotenuse

Cosine cos adjacent/ hypotenuse

Tangent tan opposite/adjacent

Cotangent cot adjacent/opposite

Secant sec hypotenuse/adjacent

Cosecant csc hypotenuse/opposite

SOHCAHTOASome old hippie cut another hippie tripping on

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Reciprocal Identities

cscθ = 1/sinθ sec θ = 1/cosθcotθ = 1/tan

tanθ = sinθ/cosθ

cotθ = cosθ/sinθ

Fundamental Identities

Sin2 θ + cos2 θ = 1

tan2 θ + 1 = sec2 θ

cot2 θ + 1 = csc2 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=5a2 + b2 = c2

32 + b2 = 52

9 + b2 = 25 b2 = 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=3a2 + b2 = c2

12 + 32 = c2

1 + 9 = c2 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 theoremCofunctions 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 TheoremExample 7b (page 399):

Find the exact value of

= = = 1

Another example:Find the exact value of

=

= = 1

Sec 5.2 HW

11-16 all25-26 all37-42 all55-60 all