Solved Name: Unit 12: Trigonometry Date: Bell: Homework 1: | Chegg.com - Free Printable

Let’s solve each problem step by step. We’ll use the Pythagorean Theorem and special right triangle rules (30-60-90 and 45-45-90) where needed.

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Problem 1:
Right triangle with legs 14 and 10, find hypotenuse x.

Use Pythagorean Theorem:
x² = 14² + 10²
x² = 196 + 100 = 296
x = √296 → simplify:
296 = 4 × 74 → so √296 = 2√74

✔ Final Answer for #1: 2√74

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Problem 2:
Right triangle with legs 7 and 16, find hypotenuse x.

x² = 7² + 16² = 49 + 256 = 305
x = √305 → can’t simplify further (305 = 5×61, no square factors)

✔ Final Answer for #2: √305

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Problem 3:
45-45-90 triangle. One leg is 2√10, other leg is x, hypotenuse is y.

In 45-45-90, legs are equal → so x = 2√10
Hypotenuse = leg × √2 → y = 2√10 × √2 = 2√20 = 2×2√5 = 4√5

✔ Final Answers for #3:
x = 2√10, y = 4√5

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Problem 4:
30-60-90 triangle. Side opposite 30° is x, side opposite 60° is 14√3, hypotenuse is y.

In 30-60-90:
- Side opposite 30° = shortest leg = let’s call it “a”
- Side opposite 60° = a√3
- Hypotenuse = 2a

We’re told side opposite 60° is 14√3 → so a√3 = 14√3 → a = 14
So x = a = 14
y = 2a = 28

✔ Final Answers for #4:
x = 14, y = 28

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Problem 5:
Triangle with angle 60°, adjacent side to 60° is y, opposite side is 30, hypotenuse is x.

This is a 30-60-90 triangle? Let’s check angles: one angle is 60°, another must be 30° since it’s a right triangle (implied by diagram).

Side opposite 60° = 30 → that’s the longer leg.

In 30-60-90:
Longer leg = a√3 → so a√3 = 30 → a = 30/√3 = 10√3
Then shorter leg (opposite 30°) = a = 10√3 → that’s y
Hypotenuse = 2a = 20√3 → that’s x

Wait — let me double-check labeling.

Diagram shows:
- Right angle at bottom left
- Angle θ = 60° at bottom right
- So side opposite 60° is vertical side = 30 → yes
- Adjacent to 60° is horizontal side = y
- Hypotenuse = x

So yes: opposite 60° = 30 = a√3 → a = 30/√3 = 10√3
Adjacent (shorter leg) = a = 10√3 → y = 10√3
Hypotenuse = 2a = 20√3 → x = 20√3

✔ Final Answers for #5:
x = 20√3, y = 10√3

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Problem 6:
Isosceles right triangle (45-45-90), base = 28, legs = x and y.

Since it’s 45-45-90, legs are equal → x = y
Base is hypotenuse = 28

In 45-45-90: hypotenuse = leg × √2 → so leg = hypotenuse / √2 = 28 / √2
Rationalize: (28√2)/2 = 14√2

So x = 14√2, y = 14√2

✔ Final Answers for #6:
x = 14√2, y = 14√2

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Problem 7:
Right triangle with legs 12 and 5, angle θ at bottom left. Find all six trig functions.

First, find hypotenuse:
hyp = √(12² + 5²) = √(144 + 25) = √169 = 13

Now, for angle θ:
- Opposite = 5
- Adjacent = 12
- Hypotenuse = 13

sinθ = opp/hyp = 5/13
cosθ = adj/hyp = 12/13
tanθ = opp/adj = 5/12
cscθ = 1/sin = 13/5
secθ = 1/cos = 13/12
cotθ = 1/tan = 12/5

✔ Final Answers for #7:
sinθ = 5/13, cosθ = 12/13, tanθ = 5/12
cscθ = 13/5, secθ = 13/12, cotθ = 12/5

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Problem 8:
Right triangle with legs 12 and 16, angle θ at top. Find all six trig functions.

First, hypotenuse: √(12² + 16²) = √(144 + 256) = √400 = 20

Angle θ is at top → so relative to θ:
- Opposite = 12 (side across from θ)
- Adjacent = 16 (side next to θ, not hypotenuse)
- Hypotenuse = 20

sinθ = opp/hyp = 12/20 = 3/5
cosθ = adj/hyp = 16/20 = 4/5
tanθ = opp/adj = 12/16 = 3/4
cscθ = 1/sin = 5/3
secθ = 1/cos = 5/4
cotθ = 1/tan = 4/3

✔ Final Answers for #8:
sinθ = 3/5, cosθ = 4/5, tanθ = 3/4
cscθ = 5/3, secθ = 5/4, cotθ = 4/3

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Final Answer Summary:

1. 2√74
2. √305
3. x = 2√10, y = 4√5
4. x = 14, y = 28
5. x = 20√3, y = 10√3
6. x = 14√2, y = 14√2
7. sinθ = 5/13, cosθ = 12/13, tanθ = 5/12, cscθ = 13/5, secθ = 13/12, cotθ = 12/5
8. sinθ = 3/5, cosθ = 4/5, tanθ = 3/4, cscθ = 5/3, secθ = 5/4, cotθ = 4/3