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
- Use the cosine rule with an included 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
Two triangle sides are 5 cm and 7 cm with included angle 60°. Find the opposite third side c.
Why this math matters
Calculate a triangle's missing side when no right angle is available. 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.
Extend Pythagoras to a nonright triangle
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Two triangle sides are 5 cm and 7 cm with included angle 60°. Find the opposite third side c.
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
c²=5²+7²−2(5)(7)cos 60°
The cosine term corrects for the angle between the known sides.
Develop the calculation
c²=25+49−70(1/2)=39
Use cos 60°=1/2 exactly.
Check and interpret
c=√39 cm≈6.245 cm
The value lies between |7−5| and 7+5, as the triangle inequality requires.
The result
c=√39 cm≈6.245 cm
The value lies between |7−5| and 7+5, as the triangle inequality requires.
Common mistakes to catch
- The angle used must be between the two sides multiplying its cosine.
- Take the positive square root for a side length.
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
What does the formula become when the included angle is 90°?
Show a hint
Cosine of ninety degrees is zero.
Reveal answer and explanation
c²=a²+b²
The correction vanishes, recovering Pythagoras.
Practice 2
For fixed positive a and b, does c grow as the included angle grows from 0° to 180°?
Show a hint
Cosine decreases over that interval.
Reveal answer and explanation
Yes
Subtracting a decreasing cosine term increases c².
Take the idea with you
Calculate a triangle's missing side when no right angle is available.
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: Use a known opposite side-angle pair
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.