What are the basic trigonometric ratios in Geometry and Trigonometry?

In the context of Geometry and Trigonometry, the basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios are derived from the sides of a right-angled triangle. Sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

How are trigonometric ratios used in the IB MYP Mathematics curriculum?

In the IB MYP Mathematics curriculum, trigonometric ratios are used to solve problems involving right-angled triangles. Students learn to apply these ratios to find unknown side lengths or angles, and they are also introduced to real-world applications such as calculating heights and distances. Understanding these ratios is fundamental for progressing to more advanced topics in trigonometry.

How do you calculate the sine, cosine, and tangent of an angle?

To calculate the sine, cosine, and tangent of an angle in a right-angled triangle, you need to know the lengths of the sides. For an angle θ, sine (sin θ) is calculated as the length of the opposite side divided by the hypotenuse, cosine (cos θ) is the adjacent side divided by the hypotenuse, and tangent (tan θ) is the opposite side divided by the adjacent side. These calculations are fundamental in trigonometry.

What is the significance of the unit circle in understanding trigonometric ratios?

The unit circle is a crucial concept in trigonometry as it provides a geometric representation of the trigonometric ratios. It is a circle with a radius of one unit centered at the origin of a coordinate plane. The unit circle helps in understanding how the sine, cosine, and tangent functions behave for different angles, especially beyond the 0 to 90-degree range, and is essential for extending trigonometric concepts to all real numbers.

Can trigonometric ratios be applied to non-right-angled triangles?

Yes, trigonometric ratios can be extended to non-right-angled triangles using the laws of sines and cosines. The Law of Sines relates the ratios of the lengths of sides to the sines of their opposite angles, while the Law of Cosines is a generalization of the Pythagorean theorem. These laws are part of the IB MYP Mathematics curriculum and are used to solve problems involving any type of triangle.

Why is "Converting exact values to decimals during the calculation." a common mistake in Trigonometric Ratios?

Why it happens: Habit. How to avoid it: Keep exact values (√(3)/2, etc.) throughout the algebra. Only convert at the end IF asked for a numerical answer.

Why is "Drawing the angle of depression from the horizontal facing the wrong way." a common mistake in Trigonometric Ratios?

Why it happens: Confusion about which side is the horizontal. How to avoid it: Always start by drawing a HORIZONTAL line from the observer's eye, then the angle of depression goes DOWN from that horizontal to the line of sight.

Geometry and TrigonometryIB MYP Maths Extended Level

Trigonometric Ratios

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Trigonometric Ratios — IB MYP Mathematics (Extended): exact values and applications

The three trig ratios extend to all angles. This MYP Mathematics Extended note covers exact values of common angles, identity relationships, and applications like elevation and depression.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Recall exact values of trig ratios.
MYP Mathematics A — Use the Pythagorean identity.
MYP Mathematics C — Apply trig to angles of elevation and depression.
MYP Mathematics D — Solve real-world height / distance problems.

Exact values for common angles

Memorise the special angles.

For $0°, 30°, 45°, 60°, 90°$ , the trig ratios have EXACT (not decimal) values you should know:

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0°$	$0$	$1$	$0$
$30°$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{3}}$
$45°$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	$1$
$60°$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$
$90°$	$1$	$0$	undefined

These come from two special triangles:

The 45-45-90 triangle with sides $1, 1, \sqrt{2}$ (an isosceles right triangle).
The 30-60-90 triangle with sides $1, \sqrt{3}, 2$ (half of an equilateral triangle).

Why exact values matter: in proofs and Extended-level work, leaving the answer as $\sqrt{3}/2$ is MORE PRECISE than $0.866$ (which is rounded). Exact values can also COMBINE algebraically, e.g. $\sqrt{3} \cdot \sqrt{3} = 3$ .

Worked example. $\sin 30° \cdot \cos 60°$ .

$= \dfrac{1}{2} \cdot \dfrac{1}{2} = \dfrac{1}{4}$ . Exact.

(Compare to decimal: $0.5 \times 0.5 = 0.25$ — same value, but less general.)

Memorise exact values for $0°, 30°, 45°, 60°, 90°$ .
From special triangles (45-45-90 and 30-60-90).
Exact values are more precise than decimals.

Trig identities

Algebraic relationships between trig ratios.

Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1.$

This holds for any angle $\theta$ . It comes from the Pythagorean theorem applied to the unit circle (next subtopic).

Quotient identity: $\tan \theta = \frac{\sin \theta}{\cos \theta}.$

(For values of $\theta$ where $\cos \theta \ne 0$ .)

Worked example. $\sin \theta = \dfrac{3}{5}$ . Find $\cos \theta$ (acute angle).

$\sin^2 + \cos^2 = 1$ .
$\dfrac{9}{25} + \cos^2 = 1$ .
$\cos^2 = \dfrac{16}{25}$ .
$\cos \theta = \dfrac{4}{5}$ (positive since $\theta$ acute).

Worked example. $\tan \theta = \dfrac{12}{5}$ . Find $\sin \theta$ (acute).

Treat as right triangle: opposite 12, adjacent 5. Hypotenuse = $\sqrt{144 + 25} = 13$ .
$\sin \theta = \dfrac{12}{13}$ .

These identities are foundational for Extended Level — they'll appear in calculus and beyond.

$\sin^2 + \cos^2 = 1$ (Pythagorean identity).
$\tan = \sin / \cos$ .
From one ratio, find the others using the identity.

Angles of elevation and depression

Real-world right-triangle problems.

An angle of elevation is the angle you look UP from the horizontal to see something above you.

An angle of depression is the angle you look DOWN from the horizontal to see something below.

Important fact: angle of elevation from A to B = angle of depression from B to A (parallel-line alternate angles!).

Worked example. Standing 50 m from the base of a tower, you measure the angle of elevation to the top as $35°$ . Find the height of the tower.

Right triangle: adjacent = 50, angle = 35°. Opposite (height) = ?
$\tan 35° = \dfrac{h}{50}$ .
$h = 50 \tan 35° \approx 50 \times 0.700 = 35.0\,\text{m}$ .

Worked example. From the top of a 100 m cliff, the angle of depression to a boat is $20°$ . How far from the base of the cliff is the boat?

Right triangle: opposite (vertical) = 100, angle = 20° (from horizontal). Adjacent (horizontal) = ?
$\tan 20° = \dfrac{100}{d}$ → $d = \dfrac{100}{\tan 20°} \approx \dfrac{100}{0.364} \approx 274.7\,\text{m}$ .

These are classic MYP criterion-D problems. Steps:

Draw a right-triangle diagram.
Identify which side is which (opposite, adjacent, hypotenuse).
Choose SOH, CAH or TOA.
Solve.

Elevation: look up. Depression: look down.
Always draw a right-triangle diagram.
Use SOHCAHTOA.

How it’s examined

Criterion A: exact values + identities. Criterion C: clear elevation/depression diagrams. Criterion D: real height/distance contexts.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Trigonometric Ratios

Step-by-step solutions to past-paper-style questions on trigonometric ratios, written exactly the way a tutor would explain them at the board.

1Use exact values
Getting started• exact
Question
Evaluate $\sin 60° + \cos 30°$ exactly.
Step-by-step solution
1. Step 1
  $\sin 60° = \dfrac{\sqrt{3}}{2}$ and $\cos 30° = \dfrac{\sqrt{3}}{2}$ .
2. Step 2
  Sum: $\dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{3}}{2} = \sqrt{3}$ .
Answer
$\sqrt{3}$
2Use the Pythagorean identity
Building confidence• identity
Question
$\cos \theta = \dfrac{5}{13}$ and $\theta$ is acute. Find $\sin \theta$ and $\tan \theta$ .
Step-by-step solution
1. Step 1
  $\sin^2 \theta = 1 - \cos^2 \theta = 1 - \dfrac{25}{169} = \dfrac{144}{169}$ .
2. Step 2
  $\sin \theta = \dfrac{12}{13}$ (positive since acute).
3. Step 3
  $\tan \theta = \dfrac{\sin}{\cos} = \dfrac{12/13}{5/13} = \dfrac{12}{5}$ .
Answer
$\sin \theta = \dfrac{12}{13}$ , $\tan \theta = \dfrac{12}{5}$ .
3Tower height (elevation)
Building confidence• elevation
Question
From 80 m away, the angle of elevation to a tower's top is $25°$ . Find the tower's height.
Step-by-step solution
1. Step 1
  Adjacent = 80, angle = $25°$ , opposite = height.
2. Step 2
  $\tan 25° = h/80 \Rightarrow h = 80\tan 25° \approx 80 \times 0.466 = 37.3\,\text{m}$ .
Answer
$\approx 37.3\,\text{m}$
4Depression angle
Stretch• depression
Question
From a 50 m lighthouse, the angle of depression to a ship is $15°$ . How far is the ship from the base of the lighthouse?
Step-by-step solution
1. Step 1
  The angle of depression from lighthouse = angle of elevation from ship (alternate angles). Right triangle: opposite = 50, angle at ship = 15°.
2. Step 2
  $\tan 15° = 50/d \Rightarrow d = 50/\tan 15° \approx 50/0.268 \approx 186.6\,\text{m}$ .
Answer
$\approx 186.6\,\text{m}$

Key Definitions and Keywords — Trigonometric Ratios

Definitions to memorise and the exact keywords mark schemes credit for trigonometric ratios answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Trigonometric ratios
Examiner keyword
Sine, cosine, and tangent — relationships between sides of a right triangle.
Pythagorean identity
Examiner keyword
$\sin^2 \theta + \cos^2 \theta = 1$ .
Angle of elevation
The angle measured upward from the horizontal to a higher object.
Angle of depression
The angle measured downward from the horizontal to a lower object.

Common Mistakes and Misconceptions — Trigonometric Ratios

The traps other students keep falling into on trigonometric ratios questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Converting exact values to decimals during the calculation.
Why it happens
Habit.
How to avoid it
Keep exact values ( $\sqrt{3}/2$ , etc.) throughout the algebra. Only convert at the end IF asked for a numerical answer.
✕Drawing the angle of depression from the horizontal facing the wrong way.
Why it happens
Confusion about which side is the horizontal.
How to avoid it
Always start by drawing a HORIZONTAL line from the observer's eye, then the angle of depression goes DOWN from that horizontal to the line of sight.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

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Trigonometric Ratios — frequently asked questions

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Trigonometric Ratios

Short Study Notes in the form of Slides

Short Study Notes — Trigonometric Ratios

Detailed Notes

What you’ll learn

How it’s examined

1Use exact values

2Use the Pythagorean identity

3Tower height (elevation)

4Depression angle

Trigonometric ratios

Pythagorean identity

Angle of elevation

Angle of depression

✕Converting exact values to decimals during the calculation.

✕Drawing the angle of depression from the horizontal facing the wrong way.

Practice questions

Past paper style quiz

Assess your understanding

Trigonometric Ratios — frequently asked questions