rigonometry links angles to side lengths, which makes it one of the most practical branches of mathematics for future seafarers. The same ideas help navigators estimate distances, heights, and bearings at sea.
In this guide, you will find the six trigonometric ratios, the special angle values worth memorizing, and step-by-step solutions to common problem types. A short quiz lets you check your progress.
A quick note: the questions and examples here were written for practice and do not come from the Philippine Merchant Marine Academy. Treat them as a way to strengthen your Mathematics foundation, not as a forecast of what the actual test will contain.
Try the Trigonometry Practice Quiz
Answer each item using a quick sketch and the formulas below. Where a problem uses a special angle such as 30°, 45°, or 60°, aim to work with exact values instead of decimals.
Loading PMMA practice quizzes…
Trigonometry at a Glance
Most trigonometry problems begin with a right triangle. Once you pick one of the acute angles, called θ (theta), each side gets a name based on its position:
- The hypotenuse is the longest side, across from the right angle.
- The opposite side lies directly across from θ.
- The adjacent side touches θ but is not the hypotenuse.
The names of the opposite and adjacent sides switch when you change the angle you are working with, so always label them from the angle in question.
The Six Trigonometric Ratios
Each ratio compares two sides of the triangle. The first three are the ones you will use most, and the mnemonic SOH-CAH-TOA keeps them straight.
| Ratio | Symbol | Formula |
|---|---|---|
| Sine | sin θ | opposite ÷ hypotenuse |
| Cosine | cos θ | adjacent ÷ hypotenuse |
| Tangent | tan θ | opposite ÷ adjacent |
| Cosecant | csc θ | hypotenuse ÷ opposite |
| Secant | sec θ | hypotenuse ÷ adjacent |
| Cotangent | cot θ | adjacent ÷ opposite |
The last three are simply the reciprocals of the first three.
Sample problem: A right triangle has sides 3, 4, and 5. If θ is opposite the side of length 3, find all six ratios.
- sin θ = 3/5, cos θ = 4/5, tan θ = 3/4
- csc θ = 5/3, sec θ = 5/4, cot θ = 4/3
Special Angle Values Worth Memorizing
Exam problems often use 0°, 30°, 45°, 60°, and 90° because their ratios have exact values. Knowing them by heart lets you skip long calculations.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Memory trick: The sine values from 0° to 90° follow the pattern √0/2, √1/2, √2/2, √3/2, √4/2. Cosine runs through the same values in reverse order. Tangent is sine divided by cosine.
Solving Right-Triangle Problems
Finding an Unknown Side
Pick the ratio that connects the side you know with the side you need, then solve for the unknown.
Problem: A gangway 10 m long rests at a 30° angle to the dock. How high above the dock is its upper end, and how far does it extend horizontally?
- The height is opposite the 30° angle, and the gangway is the hypotenuse, so use sine: height = 10 × sin 30° = 10 × 1/2 = 5 m.
- The horizontal distance is adjacent to the angle, so use cosine: distance = 10 × cos 30° = 10 × √3/2 = 5√3 m, or about 8.66 m.
Finding an Unknown Angle
When two sides are known, write the matching ratio and identify the angle that produces it.
Problem: A 14-m cable runs from the top of a 7-m mast down to the deck. What angle does the cable make with the deck?
- The mast height is opposite the angle, and the cable is the hypotenuse: sin θ = 7 ÷ 14 = 1/2.
- From the special angle table, sin 30° = 1/2, so θ = 30°.
Angles of Elevation and Depression
An angle of elevation is measured upward from a horizontal line to an object above. An angle of depression is measured downward from a horizontal line to an object below. Both are always measured from the horizontal, never from the vertical.
Because the horizontal lines at the observer and at the object are parallel, the angle of depression from one point equals the angle of elevation from the other.
Problem: From the top of a lighthouse 50 m above sea level, a keeper sees a boat at an angle of depression of 30°. How far is the boat from the base of the lighthouse?
- From the boat, the angle of elevation to the top of the lighthouse is also 30°.
- The height is opposite this angle, and the unknown distance d is adjacent, so use tangent: tan 30° = 50 ÷ d.
- Solve: d = 50 ÷ tan 30° = 50 ÷ (√3/3) = 50√3 m, or about 86.6 m.
Going Beyond Right Triangles
Degrees and Radians
Angles can be measured in degrees or radians, where 180° = π radians.
- To convert degrees to radians, multiply by π/180. For example, 60° = 60 × π/180 = π/3.
- To convert radians to degrees, multiply by 180/π. For example, 3π/4 = 3π/4 × 180/π = 135°.
Radians also give a simple formula for arc length: s = rθ, with θ in radians. On a circle with a radius of 6 cm, a central angle of π/3 cuts off an arc of 6 × π/3 = 2π cm, or about 6.28 cm.
Fundamental Identities
Identities are equations that hold true for every angle where both sides are defined. These are the ones used most:
- Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
- Quotient: tan θ = sin θ ÷ cos θ
- Pythagorean: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ
Problem: If sin θ = 3/5 and θ is acute, find cos θ and tan θ.
- Use sin²θ + cos²θ = 1: cos²θ = 1 − 9/25 = 16/25.
- Since θ is acute, cosine is positive: cos θ = 4/5.
- Divide: tan θ = (3/5) ÷ (4/5) = 3/4.
Problem: Simplify (1 − cos²θ) ÷ sin θ.
- Replace 1 − cos²θ with sin²θ: sin²θ ÷ sin θ.
- Cancel one factor: sin θ.
Signs in the Four Quadrants
For angles beyond 90°, the sign of each ratio depends on the quadrant where the angle ends. The phrase All Students Take Calculus tells you which ratios are positive, starting from Quadrant I and moving counterclockwise.
| Quadrant | Angle range | Positive ratios |
|---|---|---|
| I | 0° to 90° | All |
| II | 90° to 180° | Sine (and cosecant) |
| III | 180° to 270° | Tangent (and cotangent) |
| IV | 270° to 360° | Cosine (and secant) |
To evaluate such an angle, find its reference angle, the acute angle it makes with the x-axis, and then attach the correct sign.
- sin 150°: reference angle 30°, Quadrant II, sine positive, so sin 150° = 1/2
- cos 120°: reference angle 60°, Quadrant II, cosine negative, so cos 120° = −1/2
- tan 225°: reference angle 45°, Quadrant III, tangent positive, so tan 225° = 1
The Law of Sines and the Law of Cosines
These laws work for any triangle, not just right triangles. In each, the lowercase letter is the side opposite the angle with the matching capital letter.
- Law of Sines: a/sin A = b/sin B = c/sin C. Use it when you know two angles and a side.
- Law of Cosines: c² = a² + b² − 2ab cos C. Use it when you know two sides and the angle between them, or all three sides.
Problem: In triangle ABC, A = 30°, B = 45°, and a = 10. Find b.
- Set up the Law of Sines: b ÷ sin 45° = 10 ÷ sin 30°.
- Solve: b = 10 × (√2/2) ÷ (1/2) = 10√2, or about 14.14.
Problem: Two ships leave the same port. One sails 5 km and the other 8 km, and the angle between their courses is 60°. How far apart are they?
- Apply the Law of Cosines: c² = 5² + 8² − 2(5)(8)(cos 60°).
- Compute: c² = 25 + 64 − 80 × 1/2 = 89 − 40 = 49.
- Take the square root: the ships are 7 km apart.
Strategies That Save Time on Exam Day
- Sketch every word problem. Mark the right angle, the given angle, and the known lengths. A picture shows at once which ratio fits.
- Label sides from the angle you are using. Write O, A, and H beside the opposite, adjacent, and hypotenuse before choosing a formula.
- Reach for exact values first. If an angle is 30°, 45°, or 60°, use the special values instead of decimals. Answer choices are often written with square roots.
- Run a quick reality check. The hypotenuse must be the longest side, and sine and cosine can never be greater than 1 or less than −1.
Errors That Cost Points
- Measuring elevation or depression from the vertical. These angles always start from the horizontal line of sight.
- Swapping opposite and adjacent. This usually happens when a problem switches from one acute angle to the other. Relabel the sides every time the angle changes.
- Forgetting the sign beyond 90°. cos 120° is −1/2, not 1/2. Check the quadrant before writing the final answer.
- Distributing a ratio over a sum. sin(30° + 60°) equals sin 90°, which is 1. Adding sin 30° and sin 60° instead gives about 1.37, which is wrong.
- Using the Pythagorean theorem on a non-right triangle. For other triangles, switch to the Law of Sines or the Law of Cosines.
Keep Building Your Trigonometry Skills
Trigonometry rewards short, regular sessions more than last-minute cramming. Try redrawing the special angle table from memory each morning, then solve two or three problems that mix right triangles with the Law of Sines or Cosines.
When you get an item wrong, write down whether the slip came from labeling, choosing the ratio, or the arithmetic. Patterns in those notes show you exactly what to review next.
Looking for other subjects to cover? Our complete PMMA entrance exam reviewer and coverage guide brings together English, Mathematics, Science, and reasoning practice in one place.