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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Label sides relative to one acute angle.
- Calculate three ratios.
- Check them with a second angle.
Before you start
Fractions and recognition of a right angle.
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
A triangular model has perpendicular legs 5 cm and 12 cm and hypotenuse 13 cm. At the angle beside the 12 cm leg, find sine, cosine, and tangent.
Why this math matters
Trigonometric ratios depend on a chosen angle, not on which way a diagram faces. Clear labels prevent using an accurate formula on the wrong pair of sides.
Set up the model
A useful answer starts with clear assumptions:
- The 5 cm and 12 cm sides meet at 90°.
- θ lies between the 12 cm leg and the 13 cm hypotenuse.
- The dimensions are exact in this model.
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 the right ratio before touching a calculator
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A triangular model has perpendicular legs 5 cm and 12 cm and hypotenuse 13 cm. At the angle beside the 12 cm leg, find sine, cosine, and tangent.
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.
Identify the side roles
opposite = 5; adjacent = 12; hypotenuse = 13
The hypotenuse is across from the right angle. Of the two sides touching θ, only the non-hypotenuse side counts as adjacent.
Form the ratios
sin θ = 5/13; cos θ = 12/13; tan θ = 5/12
Numerator and denominator use the same length unit, so the units cancel. These outputs are numbers, not angles.
Check consistency
sin²θ + cos²θ = 25/169 + 144/169 = 1
The identity reflects the underlying 5² + 12² = 13² triangle relation and helps detect a swapped hypotenuse.
The result
Sine is 5/13, cosine is 12/13, and tangent is 5/12.
At the other acute angle, opposite and adjacent exchange roles. The hypotenuse remains 13 cm and never changes with the reference angle.
Common mistakes to catch
- The side touching θ is not automatically adjacent if it is the hypotenuse.
- A ratio such as 5/13 is not the angle in degrees.
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 an angle with opposite 3 and adjacent 4 in a right triangle, find tangent.
Show a hint
Use opposite divided by adjacent.
Reveal answer and explanation
3/4
The hypotenuse is not needed for tangent.
Practice 2
An acute angle has opposite side 6 and hypotenuse 10. Find cosine.
Show a hint
First find the other leg.
Reveal answer and explanation
4/5
The adjacent leg is √(100 − 36) = 8, giving cosine 8/10.
Take the idea with you
Mark the reference angle first whenever a triangle is rotated or mirrored.
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 rise and run
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