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McGraw Hill Glencoe Algebra 2, 2012 View details

1. Trigonometric Functions in Right Triangles

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Exercise 16 Page 795

Identify the hypotenuse as well as the adjacent and opposite sides of the angle. Use the Pythagorean Theorem if a side length is missing.

sin θ=2sqrt(13)/13, cos θ=3sqrt(13)/13, tan θ=2/3, csc θ=sqrt(13)/2, sec θ=sqrt(13)/3, cot θ=3/2

Practice makes perfect

In a right triangle the hypotenuse is the side that is opposite the right angle. If we take one of the acute angles as a reference, we can identify the opposite and adjacent sides to the angle.

In the given right triangle we have the opposite and the adjacent side. However, we are missing the hypotenuse.

To find the length of the hypotenuse we will substitute a= 6 and b= 9 into the Pythagorean Theorem and solve for c.

a^2+b^2=c^2

SubstituteII

a= 6, b= 9

6^2 + 9^2 = c^2

▼

Solve for c

CalcPow

Calculate power

36 + 81 = c^2

AddTerms

Add terms

117 = c^2

RearrangeEqn

Rearrange equation

c^2 = 117

SqrtEqn

sqrt(LHS)=sqrt(RHS)

c = sqrt(117)

SplitIntoFactors

Split into factors

c = sqrt(9 * 13)

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

c = 3sqrt(13)

Note that when solving the equation we only kept the principal root. This is because c is the side length of a triangle and must be positive.

Let's find the values of the six trigonometric functions for angle θ. Remember to rationalize denominators if needed.

Function	Substitute	Simplify
sin θ=opp/hyp	sin θ=6/3sqrt(13)	sin θ=2sqrt(13)/13
cos θ=adj/hyp	cos θ=9/3sqrt(13)	cos θ=3sqrt(13)/13
tan θ=opp/adj	tan θ=6/9	tan θ=2/3
csc θ=hyp/opp	csc θ=3sqrt(13)/6	csc θ=sqrt(13)/2
sec θ=hyp/adj	sec θ=3sqrt(13)/9	sec θ=sqrt(13)/3
cot θ=adj/opp	cot θ=9/6	cot θ=3/2

Showing Our Work

Rationalizing Denominators

Rationalizing a denominator means eliminating the radical expression from the denominator. We had to rationalize the denominator of the expression 63sqrt(13). To do so, we expanded the fraction by multiplying the numerator and denominator by sqrt(13).

6/3sqrt(13)

ReduceFrac

a/b=.a /3./.b /3.

2/sqrt(13)

ExpandFrac

a/b=a * sqrt(13)/b * sqrt(13)

2sqrt(13)/sqrt(13)* sqrt(13)

ProdToPowTwoFac

a* a=a^2

2sqrt(13)/(sqrt(13))^2

PowSqrt

( sqrt(a) )^2 = a