Grade 9 · Grade 9: Proof, Similarity and Trigonometry · Proof, Similarity and Right-Triangle Trigonometry
Right-Triangle Trigonometry and Coordinate Geometry
Define sine, cosine and tangent from similar triangles, then prove geometric facts with coordinates.
Learning objectives
- Define the sine, cosine and tangent ratios and explain why they depend only on the angle.
- Solve right triangles, including angles of elevation and depression.
- Compute midpoints, distances and gradients.
- Use gradient criteria for parallel and perpendicular lines.
AERO Mathematics alignment
AERO.M9.GM.3
Understand that side ratios in right triangles are properties of the angles, define the sine, cosine and tangent ratios, and use them with the Pythagorean theorem to solve right triangles.
AERO.M9.GM.4
Use coordinates to compute midpoints, distances, gradients, and to prove properties of figures, including criteria for parallel and perpendicular lines.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
Standing 40 m from a tower, you measure the angle up to the top as 32°.
Think about it
How tall is the tower, without climbing it?
Hint: Which ratio connects the opposite side to the adjacent side?
EXPLAIN
Trigonometric ratios and coordinates
All right triangles with the same acute angle are similar, so their side ratios are fixed. That is why sine, cosine and tangent depend on the angle alone.
| Ratio | Definition | Use it when you know |
|---|---|---|
| sin θ | opposite ÷ hypotenuse | Opposite and hypotenuse |
| cos θ | adjacent ÷ hypotenuse | Adjacent and hypotenuse |
| tan θ | opposite ÷ adjacent | Opposite and adjacent |
For the tower: tan 32° = h/40, so h = 40 × tan 32° ≈ 25 m. Add your eye height for the true total.
Angles of elevation and depression
The angle of elevation is measured upward from the horizontal; the angle of depression is measured downward. They are equal for the same line of sight because the horizontals are parallel.
Coordinate geometry
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2). Distance uses the Pythagorean theorem. Parallel lines have equal gradients; perpendicular lines have gradients whose product is −1.
Where this is used
Surveyors, pilots, engineers and game developers all use these ratios to turn angles into distances.
INVESTIGATION
Try it yourself
Estimate the height of a tall building using a protractor and a measured distance. Repeat from a second distance and compare your two answers, then explain the sources of error.
Watch
Basic Trigonometry
Khan Academy · 9:17
Defines sine, cosine and tangent from similar triangles.
Before you watch: Why do side ratios depend only on the angle?
- Find the missing side using tan 35° in a right triangle.
- What is the gradient of a line perpendicular to y = 2x + 1?
Interactive simulation · PhET
Trig Tour
Sweep the angle and watch sine, cosine and tangent change, noting values at 30°, 45° and 60°.
While you explore
- Why is sin 30° always 0.5, no matter the triangle size?
- What happens to tan θ as θ approaches 90°?
Key vocabulary
- Angle of elevation
- The angle you look up from level ground.
- Tangent ratio
- Opposite divided by adjacent in a right triangle.
Practice questions
0/1 correct
Level 1 · Criterion A
You stand 40 m from a tower and the angle of elevation to the top is 32°. How tall is the tower?
MYP criterion tasks
Level 2 · Criterion C
A line has gradient 2/3. Find the gradient of a perpendicular line and explain the rule.
Reflect & track
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