Trigonometry Basics: Right Triangles and the Three Core Ratios

The word "trigonometry" scares people, but the idea behind it is beautifully simple and thousands of years old: the shape of a right triangle is completely decided by its angles. If you know one angle of a right triangle and one side, you can find every other side — without measuring. That single power is how surveyors measure mountains they can't climb, how sailors found their position at sea, and how you'll solve every trigonometry question on the Civil Service Exam. This chapter turns that scary word into a friendly, reliable tool.

1. Lesson Overview

Exam trigonometry stays close to the right triangle. Almost everything comes from three ratios — sine, cosine, and tangent — captured in one memory aid, SOH-CAH-TOA. Word problems dress this up as ladders, flagpoles, shadows, and "angle of elevation" scenes, but underneath, each is a right triangle waiting for one of those three ratios.

On the Civil Service Exam, trigonometry appears in a small but predictable set of Numerical Reasoning items — usually 1 to 3 questions. It leans directly on the Geometry lesson (right triangles and the Pythagorean theorem), so the two reinforce each other.

The difficulty is intermediate. The mechanics are light once you know SOH-CAH-TOA; the exam tests whether you can correctly label opposite, adjacent, and hypotenuse relative to the given angle, recall the values for the common angles (30°, 45°, 60°, 90°), and tell elevation from depression. Master those and this becomes easy, high-confidence marks.

2. Learning Objectives

After completing this lesson you will be able to:

• State and apply the three ratios via SOH-CAH-TOA.
• Identify the opposite, adjacent, and hypotenuse sides relative to a chosen angle.
• Recall the sine, cosine, and tangent of 30°, 45°, 60°, and 90°.
• Solve for a missing side or angle in a right triangle.
• Interpret angle of elevation and angle of depression, and use their equality.
• Combine trigonometry with the Pythagorean theorem when two sides are known.
• Use the identities tan = sin/cos and sin² + cos² = 1 as checks.
• Avoid the opposite/adjacent mix-ups and elevation/depression confusion that cost marks.

3. Prerequisites

You should be comfortable with:

• Right triangles and the Pythagorean theorem (a² + b² = c²) — see the Geometry lesson.
• Fractions and division, since every ratio is one side divided by another.
• Square roots, for the exact special-angle values (√3, √2).
• Basic algebra — solving a simple equation like 0.5 = 8 ÷ x.

A quick refresher on the foundation: in a right triangle, the hypotenuse is the longest side, always opposite the right angle. The other two sides are named relative to the angle you care about — the side directly across from that angle is the opposite, and the side next to it (that isn't the hypotenuse) is the adjacent. These labels change if you switch to the other angle, which is the single most important thing to keep straight in this whole topic.

4. Why This Topic Matters

Trigonometry is the math of measuring the unreachable, and it shows up in surprisingly practical places:

• Construction and carpentry. Setting the pitch of a roof, the angle of a staircase, or the slant of a ramp for PWD access.
• Surveying and engineering. Measuring the height of a building, a tower, or a mountain from the ground using an angle.
• Navigation. Ships and planes computing distances and bearings; the angle of descent for a landing aircraft.
• Everyday estimation. Figuring how long a ladder must be to safely reach a window, or how far a flagpole's shadow stretches.
• Disaster response. Estimating the height of flood water on a slope, or the reach of a rescue crane's arm.

The whole point of trigonometry is to find a distance or height you cannot measure directly — a genuinely useful superpower that the exam rewards you for having.

5. Core Concepts

We build from the three ratios to the word-problem set-ups.

The three ratios: SOH-CAH-TOA

For a right triangle, pick one of the two non-right angles. Relative to it:

• Sine = Opposite ÷ Hypotenuse (SOH)
• Cosine = Adjacent ÷ Hypotenuse (CAH)
• Tangent = Opposite ÷ Adjacent (TOA)

That's the entire toolkit. Each ratio ties the angle to a pair of sides, so knowing the angle and one side lets you find the other.

Analogy: Think of sine, cosine, and tangent as three fixed "conversion rates" that belong to an angle. A 30° angle always has a sine of 0.5 — meaning the opposite side is always half the hypotenuse, whether the triangle is tiny or enormous. The angle sets the rate; you just plug in the side you know.

Why the ratios depend only on the angle

Here's the deep idea that makes trigonometry work: all right triangles with the same angle are the same shape, just different sizes (they're "similar"). Scaling a triangle up or down multiplies every side by the same factor, so the ratio of any two sides stays fixed. That's why "the sine of 30°" is a single number, good for every 30° right triangle in the universe. Trigonometry simply tabulates these fixed ratios.

Labeling opposite, adjacent, hypotenuse

Getting this right is 90% of the battle. Fix your chosen angle, then:

• Hypotenuse — always the side across from the right angle (longest side). It never changes.
• Opposite — the side directly across from your chosen angle.
• Adjacent — the remaining side, touching your angle (but not the hypotenuse).

Warning: If you switch to the other acute angle, "opposite" and "adjacent" swap. Always fix the angle first, then label.

Choosing which ratio to use

Look at which two sides are involved (the one you know and the one you want), and pick the ratio that connects them:

• Opposite and hypotenuse → sine.
• Adjacent and hypotenuse → cosine.
• Opposite and adjacent → tangent.

A right triangle has an angle of 30°, and the hypotenuse is 16 cm. Find the side opposite the 30° angle. Opposite and hypotenuse → sine. sin 30° = opposite ÷ 16 → opposite = 16 × 0.5 = 8 cm.

The special angles you must memorize

A handful of angles cover the vast majority of exam questions. Memorize this small table:

• 30°: sin = 0.5, cos ≈ 0.866, tan ≈ 0.577
• 45°: sin ≈ 0.707, cos ≈ 0.707, tan = 1
• 60°: sin ≈ 0.866, cos = 0.5, tan ≈ 1.732
• 90°: sin = 1, cos = 0, tan is undefined

Notice the beautiful mirror: sin 30° = cos 60°, sin 60° = cos 30°, and sin 45° = cos 45°. Sine climbs from 0 to 1 as the angle grows from 0° to 90°, while cosine falls from 1 to 0 — they're reflections. This mirror (sin θ = cos of its complement) is a great way to double-check your recall.

The exact forms are worth knowing too: sin 30° = 1/2, cos 30° = √3/2, sin 45° = cos 45° = 1/√2 (≈ 0.707), tan 45° = 1, tan 60° = √3.

Angle of elevation and angle of depression

Word problems hide right triangles behind these two terms:

• Angle of elevation — the angle you look up, measured from the horizontal, from a lower point to a higher one (ground up to a treetop).
• Angle of depression — the angle you look down, from the horizontal, from a higher point to a lower one (cliff-top down to a boat).

Crucially, for the same line of sight, the angle of elevation and the angle of depression are equal — they are alternate angles between two parallel horizontals. So if you look down at a boat at 30°, the boat looks up at you at 30° too. This lets you drop the angle into the ground-level triangle where it's easy to work with.

From the top of a 50-m cliff, the angle of depression to a boat is 30°. How far is the boat from the base? The 50-m height is opposite the 30° angle (using the equal angle of elevation at the boat); the distance to the boat is adjacent. Opposite and adjacent → tangent. tan 30° = 50 ÷ distance → distance = 50 ÷ 0.577 ≈ 86.6 m.

Trigonometry works with the Pythagorean theorem

If you already know two sides, you don't need trigonometry to find the third — the Pythagorean theorem (a² + b² = c²) does it directly. Use trig when an angle is involved; use Pythagoras when only sides are. Often a problem uses both: trig to get a second side from an angle, then Pythagoras (or a triple) to finish.

A right triangle has legs 6 and 8. The hypotenuse is √(36 + 64) = √100 = 10 (the 6-8-10 triple). Then, relative to the angle facing the side "6," sin = 6/10 = 0.6, cos = 8/10 = 0.8, tan = 6/8 = 0.75.

Two handy identities

• tan θ = sin θ ÷ cos θ. (For 30°: 0.5 ÷ 0.866 ≈ 0.577. ✓)
• sin²θ + cos²θ = 1. (For 30°: 0.5² + 0.866² = 0.25 + 0.75 = 1. ✓)

These are excellent self-checks — if your sine and cosine don't square-sum to 1, something's off.

With the ideas assembled, let's picture, tabulate, and drill.

6. Visual Learning Suggestions

• [Illustration Suggestion] The labeled triangle. A right triangle with one acute angle marked θ; label the three sides "hypotenuse" (across from the right angle), "opposite" (across from θ), and "adjacent" (touching θ). Then redraw it with the other angle marked, showing opposite and adjacent swap.
• [Illustration Suggestion] SOH-CAH-TOA fingers. Write S=O/H, C=A/H, T=O/A boxed, each with its two relevant sides highlighted, so choosing a ratio becomes "which two sides do I have?"
• [Illustration Suggestion] Elevation and depression. A cliff with a boat below; draw the horizontal line at the top (depression angle down to the boat) and the horizontal at the boat (elevation angle up to the cliff), marking them equal with matching arcs.
• [Illustration Suggestion] The special-angle table. A tidy grid of 30/45/60/90 with sin, cos, tan values, and arrows showing sine rising while cosine falls — the mirror pattern.

7. Formula Library

Special-angle values:

8. Pattern Recognition

9. Problem-Solving Framework

Step 1 — Draw the right triangle and mark the right angle and the given angle. ↓ Step 2 — Label the sides relative to the given angle: hypotenuse, opposite, adjacent. ↓ Step 3 — Match the known and wanted sides to a ratio (SOH, CAH, or TOA). ↓ Step 4 — Write the equation and solve, plugging in the special-angle value. ↓ Step 5 — Sanity-check. The hypotenuse must be the longest side; a computed side shouldn't exceed the hypotenuse; use sin² + cos² = 1 if unsure.

Why Step 2 matters most: almost every trig error is a mislabeled side. Fix the angle, label carefully, and the right ratio becomes obvious.

10. Worked Examples

Beginner

Example 1. In a right triangle, the side opposite an angle is 3 and the hypotenuse is 5. Find the sine of the angle. Solution: sin = opposite ÷ hypotenuse = 3 ÷ 5 = 0.6. Difficulty: ★☆☆☆☆

Example 2. State the tangent of a 45° angle. Solution: 1 (opposite equals adjacent at 45°). Difficulty: ★☆☆☆☆

Intermediate

Example 3 (find opposite). A right triangle has a 30° angle and a hypotenuse of 20 cm. Find the side opposite the 30° angle. Solution: sin 30° = opposite ÷ 20 → opposite = 20 × 0.5 = 10 cm. Difficulty: ★★☆☆☆

Example 4 (find adjacent). A right triangle has a 60° angle and a hypotenuse of 12 cm. Find the side adjacent to the 60° angle. Solution: cos 60° = adjacent ÷ 12 → adjacent = 12 × 0.5 = 6 cm. Difficulty: ★★☆☆☆

Example 5 (tree height). Standing 40 m from a tree, the angle of elevation to its top is 30°. How tall is the tree (ignore eye height)? Thinking: Height is opposite the 30° angle; the 40 m is adjacent → tangent. Solution: tan 30° = height ÷ 40 → height = 40 × 0.577 ≈ 23.1 m. Difficulty: ★★★☆☆

Advanced

Example 6 (ladder length). A ladder leans against a wall at a 60° angle with the ground and reaches 10 m up the wall. How long is the ladder? Thinking: The 10 m up the wall is opposite the 60° angle; the ladder is the hypotenuse → sine. Solution: sin 60° = 10 ÷ ladder → ladder = 10 ÷ 0.866 ≈ 11.55 m. Difficulty: ★★★☆☆

Example 7 (angle of depression). From a 60-m-high lighthouse, the angle of depression to a boat is 45°. How far is the boat from the base? Thinking: Elevation from the boat equals the 45° depression. Height (60) is opposite; distance is adjacent → tangent, and tan 45° = 1. Solution: tan 45° = 60 ÷ distance → distance = 60 ÷ 1 = 60 m. Difficulty: ★★★☆☆

Example 8 (all three ratios). A right triangle has legs 5 and 12. For the angle opposite the side of length 5, find sin, cos, and tan. Thinking: Hypotenuse = √(25 + 144) = 13 (the 5-12-13 triple). Solution: sin = 5/13 ≈ 0.385; cos = 12/13 ≈ 0.923; tan = 5/12 ≈ 0.417. Check: sin² + cos² = (25 + 144)/169 = 1. ✓ Difficulty: ★★★★☆

Civil Service Exam Level

Example 9 (building height with distance). The angle of elevation to the top of a building from a point 50 m away on level ground is 60°. Find the building's height. Solution: height = 50 × tan 60° = 50 × 1.732 ≈ 86.6 m. Difficulty: ★★★★☆

Example 10 (airplane). An airplane flying at an altitude of 1,000 m spots an airport at an angle of depression of 30°. Find the horizontal distance from the plane to the airport. Thinking: Altitude (1,000) is opposite the 30° angle; horizontal distance is adjacent → tangent. Solution: tan 30° = 1,000 ÷ distance → distance = 1,000 ÷ 0.577 ≈ 1,732 m. Difficulty: ★★★★☆

Example 11 (ramp). A wheelchair ramp is 8 m long and makes a 30° angle with the ground. How high does it rise? Thinking: The rise is opposite the 30° angle; the ramp is the hypotenuse → sine. Solution: rise = 8 × sin 30° = 8 × 0.5 = 4 m. Difficulty: ★★★☆☆

Example 12 (shadow and sun). A 30-m flagpole casts a shadow 30 m long. Find the angle of elevation of the sun. Thinking: Height and shadow are opposite and adjacent → tangent. tan θ = 30 ÷ 30 = 1. Solution: θ = 45° (since tan 45° = 1). Difficulty: ★★★★☆

Example 13 (complementary check). If sin θ = 0.6 in a right triangle, and the triangle's sides are whole numbers, find cos θ. Thinking: Use sin² + cos² = 1, or recognize the 3-4-5 triple. Solution: cos²θ = 1 − 0.36 = 0.64 → cos θ = 0.8 (the 3-4-5 triangle: opposite 3, hypotenuse 5, adjacent 4). Difficulty: ★★★★☆

Example 14 (two-step: trig then Pythagoras). In a right triangle, a 30° angle faces a side of 7 cm. Find the hypotenuse, then the third side. Solution: sin 30° = 7 ÷ hypotenuse → hypotenuse = 7 ÷ 0.5 = 14 cm. Third side = √(14² − 7²) = √(196 − 49) = √147 ≈ 12.12 cm. Difficulty: ★★★★★

11. Exam Tricks

• Opposite/adjacent swap. The classic trap: labeling the sides for the wrong angle. Fix the angle first, then label; if you switch angles, the labels switch too.
• Elevation vs. depression. They're equal for the same sight line, so you can always move the angle down into the ground-level triangle. Confusing the two, or thinking they differ, leads to the wrong side placement.
• Trig vs. Pythagoras. If you already have two sides, skip trig — use a² + b² = c². Reaching for sine when you don't have an angle wastes time.
• Height = distance × tan. For "height of X from a distance," the height equals the horizontal distance times the tangent of the elevation angle — a one-line shortcut.
• Special-angle recall. Memorize the 30/45/60 values cold; the exam almost always uses these, and fumbling them is the main source of lost marks.

12. Common Mistakes

• Mixing up opposite and adjacent relative to the chosen angle.
• Applying SOH-CAH-TOA to a non-right triangle, where these ratios don't directly hold.
• Forgetting the hypotenuse is the longest side, opposite the right angle — not just any long side.
• Confusing elevation with depression, or thinking they're unequal for the same line of sight.
• Using sine when tangent is needed (or vice versa) — check which two sides the problem involves.
• Misremembering special-angle values — especially swapping sin 30° (0.5) with sin 60° (0.866).

13. Shortcuts

• "Height = distance × tan θ" for elevation problems, and "reach = hypotenuse × sin θ" for ladders and ramps — memorize these ready-made forms.
• Recognize the triples inside trig. A 3-4-5 or 5-12-13 triangle gives sine/cosine/tangent instantly as simple fractions, no calculator needed.
• tan 45° = 1 turns any 45° problem into "the two legs are equal" — often solvable in one step.
• Complement mirror: if you forget cos 60°, recall it equals sin 30° = 0.5. The mirror recovers half the table.
• sin² + cos² = 1 is a fast self-check on any pair of values you compute.

14. Memory Techniques

• "SOH-CAH-TOA." Say it aloud: Sine-Opposite-Hypotenuse, Cosine-Adjacent-Hypotenuse, Tangent-Opposite-Adjacent.
• "Some Officers Have Curly Auburn Hair Till Old Age" — a classic phrase whose initials spell SOH-CAH-TOA.
• "Sine climbs, cosine falls." From 0° to 90°, sine goes 0→1 and cosine goes 1→0 — they mirror.
• "Look up to elevate, look down to depress." Elevation is upward, depression downward, and the two are equal.
• "Angle sets the rate." Sine/cosine/tangent are fixed ratios that belong to the angle, not the triangle's size.

15. Real Civil Service Exam Strategy

• Time: a direct ratio plug-in takes under 45 seconds; an elevation/depression word problem 60–90 seconds.
• Always sketch the triangle — even a rough drawing prevents the opposite/adjacent mix-up.
• Memorize the special angles, since the exam nearly always uses 30°, 45°, or 60°.
• Decide trig or Pythagoras first: angle involved → trig; only sides → Pythagoras.
• Sanity-check magnitudes: no side can exceed the hypotenuse, and sin/cos values live between 0 and 1.

16. Practice Questions

Easy

1. In a right triangle, the opposite side is 6 and the hypotenuse is 10. Find sin θ.
2. State cos 60°.
3. State tan 45°.

Medium

4. A right triangle has a 30° angle and hypotenuse 18. Find the opposite side.
5. Standing 60 m from a tower, the angle of elevation to the top is 45°. Find the tower's height.
6. A ramp 10 m long makes a 30° angle with the ground. How high does it rise?

Hard

7. From a 90-m cliff, the angle of depression to a boat is 60°. Find the boat's distance from the base.
8. A right triangle has legs 8 and 15. For the angle opposite the side of length 8, find sin, cos, and tan.

Challenge

9. The angle of elevation to the top of a building from 100 m away is 30°. Find the building's height.
10. A ladder reaches 12 m up a wall at a 60° angle with the ground. How long is the ladder, and how far is its base from the wall?

Answers and Explanations

1. 0.6. 6 ÷ 10.
2. 0.5.
3. 1.
4. 9. 18 × sin 30° = 18 × 0.5.
5. 60 m. height = 60 × tan 45° = 60 × 1.
6. 5 m. 10 × sin 30° = 10 × 0.5.
7. ≈ 51.96 m. distance = 90 ÷ tan 60° = 90 ÷ 1.732 ≈ 51.96 m.
8. sin = 8/17 ≈ 0.47, cos = 15/17 ≈ 0.88, tan = 8/15 ≈ 0.53. Hypotenuse = 17 (8-15-17 triple). Check: sin² + cos² = (64 + 225)/289 = 1. ✓
9. ≈ 57.7 m. height = 100 × tan 30° = 100 × 0.577.
10. Ladder ≈ 13.86 m; base = 12 ÷ tan 60° ≈ 6.93 m. Ladder: sin 60° = 12 ÷ ladder → ladder = 12 ÷ 0.866 ≈ 13.86 m. Base (adjacent): 12 ÷ tan 60° = 12 ÷ 1.732 ≈ 6.93 m.

17. Summary

Trigonometry links a right triangle's angle to its sides through three ratios — SOH-CAH-TOA. Fix your angle, label the opposite, adjacent, and hypotenuse, then choose the ratio that connects the side you know to the side you want. Memorize the special angles (30°, 45°, 60°, 90°) and their mirror pattern (sin θ = cos of the complement). For word problems, remember that angle of elevation equals angle of depression for the same line of sight, so you can always work in the ground-level triangle; height = distance × tan θ, and a ladder's reach = length × sin θ. When only sides are known, use the Pythagorean theorem instead. Draw the triangle, label carefully, recall the special values, and this narrow topic becomes reliable, quick points.

18. Cheat Sheet

Special angles: sin 30° = 0.5, sin 45° = 0.707, sin 60° = 0.866, sin 90° = 1 (cosine is the reverse; tan 30° = 0.577, tan 45° = 1, tan 60° = 1.732).

19. Frequently Asked Questions

How do I know whether to use sine, cosine, or tangent? Look at which two sides the problem involves. Opposite and hypotenuse → sine; adjacent and hypotenuse → cosine; opposite and adjacent → tangent. Match the sides you have and want to the right ratio.

What's the difference between angle of elevation and angle of depression? Elevation is measured upward from the horizontal (looking up at something higher); depression is measured downward (looking down at something lower). For the same line of sight, they are equal.

When do I use trigonometry versus the Pythagorean theorem? Use trigonometry when an angle is involved. Use the Pythagorean theorem (a² + b² = c²) when you know two sides and want the third — no angle needed.

Do I need a calculator for the special angles? No — memorize the 30°/45°/60°/90° values. The exam uses these almost exclusively, and they're quick to recall with the mirror pattern.

Why do the ratios stay the same no matter the triangle's size? Because all right triangles with the same angle are similar (same shape, scaled). Scaling multiplies every side equally, so the ratio between two sides is unchanged — that's the whole basis of trigonometry.

20. Mastery Checklist

• ☐ I can state SOH-CAH-TOA and what each letter means.
• ☐ I can label opposite, adjacent, and hypotenuse relative to a chosen angle.
• ☐ I have memorized the sin, cos, and tan of 30°, 45°, 60°, and 90°.
• ☐ I can solve for a missing side using the right ratio.
• ☐ I understand that angle of elevation equals angle of depression.
• ☐ I can use height = distance × tan θ and reach = hypotenuse × sin θ.
• ☐ I know when to use the Pythagorean theorem instead of trig.
• ☐ I can check my work with sin² + cos² = 1.

Tick them all and trigonometry — the art of measuring the unreachable — becomes a small but dependable set of exam points.