Right Triangle Trigonometry Notes and Worksheets - Lindsay Bowden - Free Printable

Absolutely! Let’s solve each problem step by step. We’ll use the SOH-CAH-TOA definitions:

- sin(θ) = Opposite / Hypotenuse
- cos(θ) = Adjacent / Hypotenuse
- tan(θ) = Opposite / Adjacent

We’ll go one problem at a time, identifying the sides relative to the given angle.

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Problem 1: Triangle ABC (right angle at B)

Given:
- AB = 9 (opposite to ∠C, adjacent to ∠A)
- BC = 12 (adjacent to ∠C, opposite to ∠A)
- AC = 15 (hypotenuse)

We need sin(A), cos(A), tan(A)

→ For ∠A:
- Opposite = BC = 12
- Adjacent = AB = 9
- Hypotenuse = AC = 15

So:
- sin(A) = 12/15 = 4/5
- cos(A) = 9/15 = 3/5
- tan(A) = 12/9 = 4/3

✔ Answer 1:
> sin(A) = 4/5
> cos(A) = 3/5
> tan(A) = 4/3

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Problem 2: Triangle DEF (right angle at D)

Given:
- DE = 16
- DF = 12
- EF = 20 (hypotenuse)

We need sin(F), cos(F), tan(F)

→ For ∠F:
- Opposite = DE = 16
- Adjacent = DF = 12
- Hypotenuse = EF = 20

So:
- sin(F) = 16/20 = 4/5
- cos(F) = 12/20 = 3/5
- tan(F) = 16/12 = 4/3

✔ Answer 2:
> sin(F) = 4/5
> cos(F) = 3/5
> tan(F) = 4/3

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Problem 3: Triangle QRS (right angle at R)

Given:
- QR = 5
- RS = 12
- QS = 13 (hypotenuse)

We need sin(Q), cos(Q), tan(Q)

→ For ∠Q:
- Opposite = RS = 12
- Adjacent = QR = 5
- Hypotenuse = QS = 13

So:
- sin(Q) = 12/13
- cos(Q) = 5/13
- tan(Q) = 12/5

✔ Answer 3:
> sin(Q) = 12/13
> cos(Q) = 5/13
> tan(Q) = 12/5

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Problem 4: Triangle MNO (right angle at N)

Given:
- MN = 24
- NO = 10
- MO = 26 (hypotenuse)

We need sin(M), cos(M), tan(M)

→ For ∠M:
- Opposite = NO = 10
- Adjacent = MN = 24
- Hypotenuse = MO = 26

So:
- sin(M) = 10/26 = 5/13
- cos(M) = 24/26 = 12/13
- tan(M) = 10/24 = 5/12

✔ Answer 4:
> sin(M) = 5/13
> cos(M) = 12/13
> tan(M) = 5/12

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Problem 5: Triangle TUV (right angle at U)

Given:
- TU = 3
- UV = 4
- TV = 5 (hypotenuse)

We need sin(V), cos(V), tan(V) and sin(T), cos(T), tan(T)

→ For ∠V:
- Opposite = TU = 3
- Adjacent = UV = 4
- Hypotenuse = TV = 5

→ For ∠T:
- Opposite = UV = 4
- Adjacent = TU = 3
- Hypotenuse = TV = 5

So:

For ∠V:
- sin(V) = 3/5
- cos(V) = 4/5
- tan(V) = 3/4

For ∠T:
- sin(T) = 4/5
- cos(T) = 3/5
- tan(T) = 4/3

✔ Answer 5:
> sin(V) = 3/5  sin(T) = 4/5
> cos(V) = 4/5  cos(T) = 3/5
> tan(V) = 3/4  tan(T) = 4/3

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Problem 6: Triangle GHI (right angle at H)

Given:
- GH = 9
- HI = 40
- GI = 41 (hypotenuse — check: 9² + 40² = 81 + 1600 = 1681 = 41² ✓)

We need sin(G), cos(G), tan(G) and sin(I), cos(I), tan(I)

→ For ∠G:
- Opposite = HI = 40
- Adjacent = GH = 9
- Hypotenuse = GI = 41

→ For ∠I:
- Opposite = GH = 9
- Adjacent = HI = 40
- Hypotenuse = GI = 41

So:

For ∠G:
- sin(G) = 40/41
- cos(G) = 9/41
- tan(G) = 40/9

For ∠I:
- sin(I) = 9/41
- cos(I) = 40/41
- tan(I) = 9/40

✔ Answer 6:
> sin(G) = 40/41  sin(I) = 9/41
> cos(G) = 9/41  cos(I) = 40/41
> tan(G) = 40/9  tan(I) = 9/40

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Problem 7: Triangle FGH (right angle at G)

Given:
- FG = 8
- GH = 15
- FH = 17 (hypotenuse — 8²+15²=64+225=289=17² ✓)

We need sin(F), cos(F), tan(F) and sin(H), cos(H), tan(H)

→ For ∠F:
- Opposite = GH = 15
- Adjacent = FG = 8
- Hypotenuse = FH = 17

→ For ∠H:
- Opposite = FG = 8
- Adjacent = GH = 15
- Hypotenuse = FH = 17

So:

For ∠F:
- sin(F) = 15/17
- cos(F) = 8/17
- tan(F) = 15/8

For ∠H:
- sin(H) = 8/17
- cos(H) = 15/17
- tan(H) = 8/15

✔ Answer 7:
> sin(F) = 15/17  sin(H) = 8/17
> cos(F) = 8/17  cos(H) = 15/17
> tan(F) = 15/8  tan(H) = 8/15

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Problem 8: Triangle RST (right angle at S)

Given:
- RS = 12
- ST = 35
- RT = 37 (hypotenuse — 12²+35²=144+1225=1369=37² ✓)

We need sin(R), cos(R), tan(R) and sin(T), cos(T), tan(T)

→ For ∠R:
- Opposite = ST = 35
- Adjacent = RS = 12
- Hypotenuse = RT = 37

→ For ∠T:
- Opposite = RS = 12
- Adjacent = ST = 35
- Hypotenuse = RT = 37

So:

For ∠R:
- sin(R) = 35/37
- cos(R) = 12/37
- tan(R) = 35/12

For ∠T:
- sin(T) = 12/37
- cos(T) = 35/37
- tan(T) = 12/35

✔ Answer 8:
> sin(R) = 35/37  sin(T) = 12/37
> cos(R) = 12/37  cos(T) = 35/37
> tan(R) = 35/12  tan(T) = 12/35

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## ✔ Final Answers Summary:

1.
sin(A) = 4/5
cos(A) = 3/5
tan(A) = 4/3

2.
sin(F) = 4/5
cos(F) = 3/5
tan(F) = 4/3

3.
sin(Q) = 12/13
cos(Q) = 5/13
tan(Q) = 12/5

4.
sin(M) = 5/13
cos(M) = 12/13
tan(M) = 5/12

5.
sin(V) = 3/5, sin(T) = 4/5
cos(V) = 4/5, cos(T) = 3/5
tan(V) = 3/4, tan(T) = 4/3

6.
sin(G) = 40/41, sin(I) = 9/41
cos(G) = 9/41, cos(I) = 40/41
tan(G) = 40/9, tan(I) = 9/40

7.
sin(F) = 15/17, sin(H) = 8/17
cos(F) = 8/17, cos(H) = 15/17
tan(F) = 15/8, tan(H) = 8/15

8.
sin(R) = 35/37, sin(T) = 12/37
cos(R) = 12/37, cos(T) = 35/37
tan(R) = 35/12, tan(T) = 12/35

All fractions are reduced! 🎯