Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 10 chapters and video availability01 · Read and understand
What you will learn
- Match a right-triangle ratio to a chosen reference angle.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Algebraic equations, ratios, angle and area facts, and basic coordinate geometry.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For an acute angle θ in a right triangle, the opposite side is 5 and the hypotenuse is 13. Find sin θ, cos θ, and tan θ.
Why this math matters
Choose a ratio from the actual known and needed sides instead of selecting a formula at random. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use Euclidean geometry and the angle, parallelism, similarity, or congruence conditions stated; a sketch alone does not establish them.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Choose sine, cosine, or tangent from the known sides
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For an acute angle θ in a right triangle, the opposite side is 5 and the hypotenuse is 13. Find sin θ, cos θ, and tan θ.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
Adjacent side = √(13²−5²)=12
Pythagoras recovers the missing leg.
Develop the calculation
sin θ=5/13; cos θ=12/13
Sine uses opposite over hypotenuse; cosine uses adjacent over hypotenuse.
Check and interpret
tan θ=5/12
Tangent compares the two legs, and opposite/adjacent depends on the chosen angle.
The result
tan θ=5/12
Tangent compares the two legs, and opposite/adjacent depends on the chosen angle.
Common mistakes to catch
- Label the reference angle before naming opposite and adjacent.
- The hypotenuse is always opposite the right angle.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
For the other acute angle, what is sine?
Show a hint
The twelve-unit leg becomes opposite.
Reveal answer and explanation
12/13
Changing the reference angle swaps the roles of the legs.
Practice 2
Can the adjacent side in a cosine ratio be the hypotenuse?
Show a hint
The named adjacent leg excludes the hypotenuse.
Reveal answer and explanation
No
Although both touch the angle, cosine's numerator is the non-hypotenuse side.
Take the idea with you
Choose a ratio from the actual known and needed sides instead of selecting a formula at random.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Recover an angle from a measured ratio
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