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In this section,

we will be discussing how to solve real-world trigonometric equations. The fields of medicine, engineering, and computer science all use trigonometric equations. This may be hard to believe at first, but the more you look, the more you’ll find trigonometry in reality. Take a look:

These equations are solved following the same steps as simple trigonometric equations in section 4.1 (4.1 solving simple trigonometric equations):

  • Isolate the trigonometric function.
  • Determine the quadrants in which the reference angle will land in.
  • Find the reference angle.
  • Calculate the solution(s).

Example:

y = 23 + cos ((π/4 ⋅ x) - 3π/4)

Graphically determine the first two times you were 35 feet off the ground

35 = 23 + cos ((π/4 ⋅ x) - 3π/4)
-23

12 = 20cosμ
20 20

3 = cosμ
5

It is quadrants I and IV← Locate the quadrants

μ’ = cos’ (⅗) ← Then find the reference angle
μ = .927
μ = .927 or μ =5.35

Now, we have to sub μ back in to solve for x

((π/4 ⋅ x) - 3π/4) = .927 ←Sub u back in for x ((π/4 ⋅ x) - 3π/4) = 5.35 ← Sub u back in for x
3π/4 + 3π/4 + 3π/4 +3π/4

(π/4 ⋅ x) = 3.28 (π/4 ⋅ x) = 7.71
π/4 π/4 π/4 π/4
x = 4.18 x = 9.82

Solution: {4.18, 9.82}

Practice problem:

ANSWER KEY

1) 136 = 140 + 14cos((π/20⋅ x) - 9π/5)