Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Form a quotient identity and retain denominator restrictions.
- Justify the conclusion "tan15°=2−√3" using the stated assumptions.
Before you start
Angle-addition formulas and tangent.
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
Find tan15° exactly using 45°−30°.
Why this math matters
Form a quotient identity and retain denominator restrictions. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The specific angles have nonzero cosines.
- The tangent difference is defined.
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.
Derive a tangent difference from sine and cosine
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find tan15° exactly using 45°−30°.
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.
Build the model
tan(a−b)=(tan a−tan b)/(1+tan a tan b)
Divide the sine and cosine subtraction formulas by cos a cos b where those factors are nonzero.
Work through the mathematics
tan15°=(1−1/√3)/(1+1/√3)
Substitute the two known tangent values.
Check the conclusion
tan15°=2−√3
Simplifying (√3−1)/(√3+1) by a conjugate gives the exact result.
The result
tan15°=2−√3
Simplifying (√3−1)/(√3+1) by a conjugate gives the exact result.
Common mistakes to catch
- The denominator uses a plus sign for a difference angle.
- An identity derived by division retains the divided factors' restrictions.
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
Why must the final denominator be nonzero?
Show a hint
Tangent is undefined where cosine vanishes.
Reveal answer and explanation
A zero denominator corresponds to an excluded difference angle
A symbolic quotient cannot bypass tangent's domain.
Practice 2
Check the sign of 2−√3.
Show a hint
Compare √3 with two.
Reveal answer and explanation
It is positive
Fifteen degrees lies in the first quadrant.
Take the idea with you
Recover a small angular difference from two known directional slopes.
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: Diagnose a jump through unequal one-sided limits
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