Integrated Math II Name: ________________________________________ Period: ______ Date: _______________

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4.2.3 Inverse Trigonometry 4-80. Examine the triangles below. Note: The diagrams are not drawn to scale. With your partner: Identify which tool to use based on where the reference angle (θ) is located and which side lengths are involved. Write and solve an equation to determine the missing side length or angle.

a)

𝑐𝑜𝑠𝜃 =1224

𝑐𝑜𝑠!!(𝑐𝑜𝑠𝜃) = 𝑐𝑜𝑠!!(1224)

𝜃 = 𝑐𝑜𝑠!!1224

𝜃 ≈ 60°

b)

𝑡𝑎𝑛𝜃 =88

𝑡𝑎𝑛!!(𝑡𝑎𝑛𝜃) = 𝑡𝑎𝑛!!(88)

𝜃 = 𝑡𝑎𝑛!!88

𝜃 ≈ 45°

c)

𝑠𝑖𝑛57 =7𝑥

𝑥𝑠𝑖𝑛57 = 7

𝑥 =7

𝑠𝑖𝑛57

𝑥 ≈ 8.35

d) 𝑡𝑎𝑛𝑥 = !

!

𝑡𝑎𝑛!!(𝑡𝑎𝑛𝑥) = 𝑡𝑎𝑛!!(57)

𝑥 = 𝑡𝑎𝑛!!57

𝑥 ≈ 36°

e) 𝑡𝑎𝑛𝑢 =

77

𝑡𝑎𝑛!!(𝑡𝑎𝑛𝑢) = 𝑡𝑎𝑛!!(79)

𝑢 = 𝑡𝑎𝑛!!79

𝑢 ≈ 38°

f) 𝑡𝑎𝑛𝑦 =

1218

𝑡𝑎𝑛!!(𝑡𝑎𝑛𝑦) = 𝑡𝑎𝑛!!(1218)

𝑦 = 𝑡𝑎𝑛!!1718

𝑦 ≈ 43°

g) 𝑡𝑎𝑛𝑣 = !"

!!

𝑡𝑎𝑛!!(𝑡𝑎𝑛𝑣) = 𝑡𝑎𝑛!!(7888)

𝑣 = 𝑡𝑎𝑛!!7888

𝑣 ≈ 42°

h) 𝑡𝑎𝑛𝜃 = !

!"

𝑡𝑎𝑛!!(𝑡𝑎𝑛𝜃) = 𝑡𝑎𝑛!!(714)

𝜃 = 𝑡𝑎𝑛!!714

𝜃 ≈ 27°

i) 𝑡𝑎𝑛𝜃 = !"

!"

𝑡𝑎𝑛!!(𝑡𝑎𝑛𝜃) = 𝑡𝑎𝑛!!(4230)

𝜃 = 𝑡𝑎𝑛!!4230

𝜃 ≈ 54°

Integrated Math II Name: ________________________________________ Period: ______ Date: _______________

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4-78. Mr. Gow needs to build a wheelchair access ramp for the school’s auditorium. The ramp must rise a total of 3 feet to get from the ground to the entrance of the building. In order to follow the state building code, the angle formed by the ramp and the ground cannot exceed 4.76°. Mr. Gow has plans from the planning department that call for the ramp to start 25 feet away from the building. Will this ramp meet the state building code?

a) Draw an appropriate diagram. Add all the measurements you can. What does Mr. Gow need to know?

He needs to know the measure of the angle the ramp makes with the ground.

b) According to the plan, what angle will the ramp make with the ground? Justify your answer. Will the ramp be to code?

The angle will be about 6.85° because 𝑡𝑎𝑛!! !!"

≈ 6.85°. The ramp will not be to code.

c) At least how far from the building must the ramp start in order to meet the building code? If Mr. Gow builds the ramp exactly to code, how long will the ramp be? Show all work.

The ramp must be about 36.0028 feet way from the building since 𝑡𝑎𝑛 4.76° = !! ; x ≈ 36.028. The

ramp will be 36.152 ft long.

4-79. For the triangle at right, determine the measures of ∠A and ∠B. Once you have solved for the measure of the first acute angle (either ∠A or ∠B), what knowledge about the angles in triangles could help you solve for the second acute angle measure?

m∠A ≈ 21.0º, m∠B ≈ 69.0º; Triangle Angle Sum Theorem (interior angles sum =180) or Complementary angles (sum =90) 4-81. Peter cannot figure out what he has done wrong. He wrote the equation 𝑐𝑜𝑠𝜃 = !

!.!"# solve for

the missing angle of a triangle. However, his calculator gives him an error message each time he tries to calculate the angle. Jeri, his teammate, looks at his work and exclaims, “Of course your calculator gives you an error! Your cosine ratio is impossible!” What is Jeri talking about? And how can she tell without seeing the triangle or checking on her calculator? Since cosine is the ratio of the adjacent side to the hypotenuse, and since the hypotenuse is the longest side of a right triangle, the denominator is always bigger than (or equal to, in the case of 0°) the numerator. Thus, the value of cosine is always ≤ 1.

Integrated Math II Name: ________________________________________ Period: ______ Date: _______________

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