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 12 chapters and video availability01 · Read and understand
What you will learn
- Separate algebraic roots from angle solutions and range checks.
- Justify the conclusion "x=7π/6,11π/6,π/2" using the stated assumptions.
Before you start
Factoring and the unit circle.
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
Solve 2sin²x−sin x−1=0 on [0,2π).
Why this math matters
Separate algebraic roots from angle solutions and range checks. 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 requested interval includes zero and excludes 2π.
- Angles use radians.
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.
Solve a polynomial equation in a trigonometric value
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve 2sin²x−sin x−1=0 on [0,2π).
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
Let s=sin x; 2s²−s−1=(2s+1)(s−1)
Treat the trigonometric value as an algebraic unknown first.
Work through the mathematics
s=−1/2 or s=1, both inside [−1,1]
Algebraic candidates must fit sine's possible range.
Check the conclusion
x=7π/6,11π/6,π/2
Recover every angle in the specified interval for both allowed sine values.
The result
x=7π/6,11π/6,π/2
Recover every angle in the specified interval for both allowed sine values.
Common mistakes to catch
- A root of the algebraic substitution is not yet an angle.
- Do not omit the second angle for a nonextreme sine value.
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
Would an algebraic candidate sin x=2 give real angles?
Show a hint
Compare with sine's range.
Reveal answer and explanation
No
Such a root must be rejected at the angle-recovery stage.
Practice 2
How many solutions lie in [0,4π)?
Show a hint
Repeat the three solutions one period later.
Reveal answer and explanation
Six
None of the displayed solutions is a duplicated endpoint.
Take the idea with you
Use a substitution to separate algebraic solving from periodic angle bookkeeping.
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: Derive a tangent difference from sine and cosine
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