Trigonometric Ratios
📖 Learn
Trigonometry is the study of relationships between angles and sides in triangles. In this lesson, we focus on right triangle trigonometry and the three primary trigonometric ratios.
Parts of a Right Triangle
Every right triangle has:
- Hypotenuse: The longest side, opposite the right angle
- Opposite side: The side across from the reference angle (not the right angle)
- Adjacent side: The side next to the reference angle (not the hypotenuse)
The terms "opposite" and "adjacent" depend on which acute angle you're considering.
The Three Primary Trigonometric Ratios
For an acute angle θ in a right triangle:
| Ratio | Abbreviation | Formula |
|---|---|---|
| Sine | sin θ | Opposite / Hypotenuse |
| Cosine | cos θ | Adjacent / Hypotenuse |
| Tangent | tan θ | Opposite / Adjacent |
This mnemonic helps remember the ratios:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
Special Right Triangles
Two special right triangles have exact trigonometric ratios that you should memorize:
| Triangle | Sides | sin | cos | tan |
|---|---|---|---|---|
| 45-45-90 | 1 : 1 : √2 | sin 45° = √2/2 | cos 45° = √2/2 | tan 45° = 1 |
| 30-60-90 | 1 : √3 : 2 | sin 30° = 1/2 sin 60° = √3/2 |
cos 30° = √3/2 cos 60° = 1/2 |
tan 30° = √3/3 tan 60° = √3 |
Reciprocal Trigonometric Ratios
| Ratio | Abbreviation | Formula | Reciprocal Of |
|---|---|---|---|
| Cosecant | csc θ | Hypotenuse / Opposite | 1 / sin θ |
| Secant | sec θ | Hypotenuse / Adjacent | 1 / cos θ |
| Cotangent | cot θ | Adjacent / Opposite | 1 / tan θ |
💡 Examples
Example 1: Finding Trigonometric Ratios
Problem: In a right triangle, the sides are 5, 12, and 13. Find sin A, cos A, and tan A where A is the angle opposite the side of length 5.
Solution:
First, identify the sides relative to angle A:
Opposite = 5, Adjacent = 12, Hypotenuse = 13
sin A = Opposite/Hypotenuse = 5/13
cos A = Adjacent/Hypotenuse = 12/13
tan A = Opposite/Adjacent = 5/12
Answer: sin A = 5/13, cos A = 12/13, tan A = 5/12
Example 2: Using Trig to Find a Missing Side
Problem: In a right triangle, one angle is 35° and the hypotenuse is 10. Find the length of the side opposite the 35° angle.
Solution:
We know the angle and hypotenuse, and want the opposite side.
Use sine: sin 35° = Opposite/Hypotenuse
sin 35° = x/10
x = 10 × sin 35°
x = 10 × 0.5736 ≈ 5.74
Answer: The opposite side is approximately 5.74 units
Example 3: Using Special Triangles
Problem: Find the exact value of sin 60° + cos 30°.
Solution:
From the 30-60-90 triangle:
sin 60° = √3/2
cos 30° = √3/2
sin 60° + cos 30° = √3/2 + √3/2 = 2√3/2 = √3
Answer: √3
Example 4: Finding an Adjacent Side
Problem: A ladder leans against a wall at a 70° angle with the ground. If the ladder is 15 feet long, how far is the base of the ladder from the wall?
Solution:
The ladder is the hypotenuse (15 ft), and we want the distance from the wall (adjacent to the 70° angle).
cos 70° = Adjacent/Hypotenuse
cos 70° = x/15
x = 15 × cos 70°
x = 15 × 0.342 ≈ 5.13
Answer: The base is approximately 5.13 feet from the wall
Example 5: Using Tangent
Problem: From a point 50 meters from the base of a building, the angle of elevation to the top is 62°. How tall is the building?
Solution:
The distance from the point to the building is the adjacent side (50 m).
The height of the building is the opposite side.
tan 62° = Opposite/Adjacent
tan 62° = h/50
h = 50 × tan 62°
h = 50 × 1.881 ≈ 94.05
Answer: The building is approximately 94 meters tall
✏️ Practice
Try these problems on your own. Choose the best answer for each question.
1. In a right triangle with legs 8 and 15 and hypotenuse 17, what is sin A if angle A is opposite the leg of length 8?
A) 8/17
B) 15/17
C) 8/15
D) 17/8
2. What is the exact value of tan 45°?
A) 0
B) 1/2
C) 1
D) √2
3. If cos θ = 3/5, what is sin θ in the same right triangle?
A) 3/5
B) 4/5
C) 5/3
D) 4/3
4. Which ratio equals the hypotenuse divided by the opposite side?
A) sin θ
B) cos θ
C) csc θ
D) tan θ
5. What is the exact value of sin 30°?
A) 1/2
B) √2/2
C) √3/2
D) 1
6. A right triangle has an angle of 40° and the adjacent side is 12. What equation would you use to find the opposite side?
A) sin 40° = x/12
B) cos 40° = x/12
C) tan 40° = x/12
D) tan 40° = 12/x
7. In a 45-45-90 triangle with legs of length 6, what is the length of the hypotenuse?
A) 6
B) 6√2
C) 12
D) 3√2
8. If tan θ = 3/4, what is cot θ?
A) 3/4
B) 4/3
C) 4/5
D) 3/5
9. What is cos 60°?
A) 1/2
B) √2/2
C) √3/2
D) √3/3
10. A guy wire is attached to a pole 20 feet above the ground. If the wire makes a 55° angle with the ground, approximately how long is the wire?
A) 20 × sin 55°
B) 20/sin 55°
C) 20 × cos 55°
D) 20/cos 55°
Click to reveal answers
- A) 8/17 — sin = Opposite/Hypotenuse = 8/17
- C) 1 — In a 45-45-90 triangle, opposite = adjacent, so tan 45° = 1
- B) 4/5 — If cos θ = 3/5, then adj = 3, hyp = 5, opp = 4 (by Pythagorean), so sin θ = 4/5
- C) csc θ — Cosecant is the reciprocal of sine: csc θ = hyp/opp
- A) 1/2 — From the 30-60-90 triangle, sin 30° = 1/2
- C) tan 40° = x/12 — tan = opp/adj, so tan 40° = x/12
- B) 6√2 — In a 45-45-90 triangle, hypotenuse = leg × √2
- B) 4/3 — cot θ = 1/tan θ = 1/(3/4) = 4/3
- A) 1/2 — From the 30-60-90 triangle, cos 60° = 1/2
- B) 20/sin 55° — sin 55° = 20/wire, so wire = 20/sin 55°
✅ Check Your Understanding
Question 1: Why is the hypotenuse always in the denominator for sine and cosine, but not tangent?
Reveal Answer
Sine and cosine are defined as ratios compared to the hypotenuse: sin = opp/hyp and cos = adj/hyp. This makes their values always between -1 and 1. Tangent, however, compares the two legs directly (tan = opp/adj), which represents the slope or steepness of the angle. Tangent can have any value from negative infinity to positive infinity. The hypotenuse is the reference for "normalizing" sine and cosine.
Question 2: How do you decide which trigonometric ratio to use when solving a problem?
Reveal Answer
Identify which sides you know and which you need to find, relative to the given angle. If you have or need the hypotenuse and the opposite side, use sine. If you have or need the hypotenuse and adjacent side, use cosine. If you're working with only the two legs (opposite and adjacent), use tangent. Remember: pick the ratio that involves the sides you know and need.
Question 3: Why are the values for sin 30° and cos 60° both equal to 1/2?
Reveal Answer
In a 30-60-90 triangle, the side opposite 30° is the same as the side adjacent to 60°. Both are the shorter leg. So sin 30° = (short leg)/hypotenuse and cos 60° = (short leg)/hypotenuse give the same ratio. This relationship is called a cofunction identity: sin θ = cos(90° - θ). The sine of an angle equals the cosine of its complement.
Question 4: For what angle is sin θ = cos θ, and why?
Reveal Answer
sin θ = cos θ when θ = 45°. In a 45-45-90 triangle, the two legs are equal in length (it's an isosceles right triangle). Since the opposite and adjacent sides are the same, the ratios sin 45° = opp/hyp and cos 45° = adj/hyp are equal. Both equal √2/2, or equivalently, 1/√2.
🚀 Next Steps
- Review any concepts that felt challenging
- Move on to the next lesson when ready
- Return to practice problems periodically for review