Trigonometric Ratios
Sine, Cosine, and Tangent in Right-Angled Triangles
What is Trigonometry?
Trigonometry is the study of the relationships between the sides and angles of triangles. The three primary trigonometric ratios — sine, cosine, and tangent — relate an angle to the ratios of two sides in a right-angled triangle.
Remember the ratios with:
SOH CAH TOA
The Three Ratios
Sine (sin)
SOH
\[\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\]
"Some Old Horses"
Cosine (cos)
CAH
\[\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\]
"Can Always Hear"
Tangent (tan)
TOA
\[\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\]
"Their Owner Approach"
Identifying the Sides
Hypotenuse: The longest side, opposite the right angle (always the same)
Opposite: The side across from the angle you're working with
Adjacent: The side next to the angle (not the hypotenuse)
Finding a Side
Example 1: Finding the Opposite Side
In a right-angled triangle, one angle is 35° and the hypotenuse is 10 cm. Find the opposite side.
Angle = 35°, Hypotenuse = 10 cm, Opposite = ?
We have Opposite and Hypotenuse → use sin
\[\sin 35° = \frac{\text{Opposite}}{10}\]
\[\text{Opposite} = 10 \times \sin 35° = 10 \times 0.574 = 5.74 \text{ cm}\]
Example 2: Finding the Adjacent Side
A ladder makes an angle of 70° with the ground and reaches 8 m up a wall. How far is the base of the ladder from the wall?
Angle = 70°, Opposite (height) = 8 m, Adjacent (base) = ?
We have Opposite and want Adjacent → use tan
\[\tan 70° = \frac{8}{\text{Adjacent}}\]
\[\text{Adjacent} = \frac{8}{\tan 70°} = \frac{8}{2.747} = 2.91 \text{ m}\]
Finding an Angle
Example 3: Finding an Angle
A ramp is 5 m long and rises 1.5 m. Find the angle of elevation.
Hypotenuse (ramp) = 5 m, Opposite (rise) = 1.5 m
\[\sin\theta = \frac{1.5}{5} = 0.3\]
\[\theta = \sin^{-1}(0.3) = 17.5°\]
Interactive Trigonometry Calculator
Calculate Trig Ratios
sin θ
0.707
cos θ
0.707
tan θ
1.000
For a hypotenuse of 1: Opposite = 0.707, Adjacent = 0.707
Special Angles
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | \(\frac{1}{2}\) = 0.5 | \(\frac{\sqrt{3}}{2}\) ≈ 0.866 | \(\frac{1}{\sqrt{3}}\) ≈ 0.577 |
| 45° | \(\frac{\sqrt{2}}{2}\) ≈ 0.707 | \(\frac{\sqrt{2}}{2}\) ≈ 0.707 | 1 |
| 60° | \(\frac{\sqrt{3}}{2}\) ≈ 0.866 | \(\frac{1}{2}\) = 0.5 | \(\sqrt{3}\) ≈ 1.732 |
Angles of Elevation and Depression
Key Terms
Angle of Elevation: The angle measured upward from the horizontal to a line of sight
Angle of Depression: The angle measured downward from the horizontal to a line of sight
Example 4: Angle of Elevation
A person standing 50 m from a building looks up at an angle of 40° to see the top. How tall is the building? (Assume eye level is 1.6 m)
\[\tan 40° = \frac{h}{50}\]
\[h = 50 \times \tan 40° = 50 \times 0.839 = 41.95 \text{ m}\]
Total height = 41.95 + 1.6 = 43.55 m
Practice Problems
Question 1: In a right triangle with angle 50° and adjacent side 8 cm, find the opposite side.
Show Solution
\(\tan 50° = \frac{\text{Opp}}{8}\)
\(\text{Opp} = 8 \times \tan 50° = 8 \times 1.192 = 9.54\) cm
Question 2: Find angle θ if sin θ = 0.6
Show Solution
\(\theta = \sin^{-1}(0.6) = 36.87° \approx 36.9°\)
Question 3: A kite string is 80 m long and makes an angle of 55° with the ground. How high is the kite?
Show Solution
String = hypotenuse = 80 m, height = opposite
\(\sin 55° = \frac{h}{80}\)
\(h = 80 \times \sin 55° = 80 \times 0.819 = 65.5\) m
Question 4: From the top of a 25 m tower, the angle of depression to a car is 32°. How far is the car from the base of the tower?
Show Solution
Angle of depression = 32°, so angle at ground = 32°
Opposite = 25 m, Adjacent = distance = ?
\(\tan 32° = \frac{25}{d}\)
\(d = \frac{25}{\tan 32°} = \frac{25}{0.625} = 40\) m
CSEC Exam Tips
- Always draw and label a diagram
- Identify Opposite, Adjacent, Hypotenuse from the given angle
- Use SOH CAH TOA to choose the correct ratio
- Make sure your calculator is in DEGREE mode
- For finding angles, use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹)
- Check: angles should be between 0° and 90° in right triangles