Foundations path · F10

Before this lesson: Powers, radicals and logarithms, Geometric language and ratios

Your goal: Use right-triangle ratios, angle units and trigonometric identities with valid domains.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Trigonometry foundations: the key idea

In a right triangle, sine and cosine of an acute angle compare the opposite and adjacent sides with the hypotenuse; tangent compares opposite with adjacent. Scaling the triangle leaves these ratios unchanged. The unit-circle definitions extend sine and cosine to other angles, where signs matter. Degrees and radians are different units: 180°=π radians. The identity sin²θ+cos²θ=1 follows from Pythagoras. A reciprocal or quotient identity may be used only where its denominator is nonzero.

Right triangle with legs 3 and 4 and hypotenuse 5.ABC345
A 3–4–5 right triangle. The right-angle marker identifies the two legs.

A worked example

In a right triangle, the sides opposite and adjacent to angle θ are 3 and 4. Find sin θ and cos θ.

The hypotenuse is 32+42=5\sqrt{3^{2}+4^{2}}=5. Therefore sin θ=3/5 and cos θ=4/5. Their squares sum to 9/25+16/25=1.

Your turn: change one thing

For an acute θ with sin θ=5/13, find cos θ.

Try this on paper before opening the explanation.

Compare your reasoning

The identity gives cos²θ=1−25/169=144/169. Because θ is acute, cosine is positive, so cos θ=12/13.

Pause and check

A trap to avoid: State the domain, keep exact values and explain why each step is allowed. A correct answer still needs a reason.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove that tan θ=sin θ/cos θ whenever cos θ≠0.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Using unit-circle coordinates (x,y)=(cos θ,sin θ), tangent is the slope y/x whenever x≠0. Thus tan θ=sin θ/cos θ. For an acute right-triangle angle, this also follows by cancelling the common hypotenuse in (opposite/hypotenuse)/(adjacent/hypotenuse).

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. In a right triangle, sin θ equals which ratio?

Foundation

  1. Hypotenuse / opposite
  2. Opposite / adjacent
  3. Opposite / hypotenuse
  4. Adjacent / hypotenuse
Hint

Identify the definition, not the reciprocal.

Answer and reasoning

Opposite / hypotenuse. Sine compares the opposite side with the hypotenuse.

2. How many degrees are π radians?

Foundation

  1. 60
  2. 180
  3. 90
  4. 360
Hint

Recall one straight angle.

Answer and reasoning

180. A straight angle measures both 180° and π radians.

3. For acute θ, sin θ=3/5. Find cos θ.

Core

  1. −4/5
  2. 3/5
  3. 2/5
  4. 4/5
Hint

Use the square identity and the sign.

Answer and reasoning

4/5. cos²θ=1−9/25=16/25. Acuteness selects +4/5.

4. What is sin²θ+cos²θ?

Core

  1. 0
  2. 2
  3. sin θ+cos θ
  4. 1
Hint

Use the unit-circle equation.

Answer and reasoning

1. The coordinates of a point on the unit circle satisfy x²+y²=1.

5. What restriction is needed for tan θ=sin θ/cos θ?

Stretch

  1. θ must be acute
  2. cos θ≠0
  3. sin θ≠0
  4. θ must be 0°
Hint

A quotient needs a nonzero denominator.

Answer and reasoning

cos θ≠0. The identity is valid whenever cosine is nonzero, not only for acute angles.

6. If 0°<θ<180° and sin θ=1/2, what are all possibilities?

Stretch

  1. 30° and 330°
  2. 30° and 150°
  3. Only 30°
  4. Only 150°
Hint

Sine agrees at supplementary angles.

Answer and reasoning

30° and 150°. The angles in the stated interval are 30° and 180°−30°=150°. The interval excludes 330°.

7. In a right triangle, sine of an acute angle is which ratio?

Foundation

  1. Adjacent leg / hypotenuse
  2. Hypotenuse / opposite leg
  3. Opposite leg / adjacent leg
  4. Opposite leg / hypotenuse
Hint

Use SOH in the standard ratios.

Answer and reasoning

Opposite leg / hypotenuse. Sine compares the opposite leg to the hypotenuse.

8. An acute angle has opposite leg 5 and adjacent leg 12. Its sine?

Core

  1. 5/13
  2. 5/12
  3. 12/13
  4. 13/5
Hint

Find the hypotenuse first.

Answer and reasoning

5/13. The hypotenuse is 25+144=13\sqrt{25+144}=13, so sine is 5/13.

9. Why must degree and radian units not be mixed?

Stretch

  1. The same numerical input can represent different angles
  2. Sine is defined only in degrees
  3. Radians have no conversion
  4. Every angle equals its sine
Hint

180° equals π radians.

Answer and reasoning

The same numerical input can represent different angles. A formula or calculator must interpret all angle inputs consistently.

Choose your next step

Try a written problem in the challenge room. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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