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 11 chapters and video availability01 · Read and understand
What you will learn
- Combine reference-angle solutions with full-period shifts.
- Justify the conclusion "x=π/6+2kπ or x=5π/6+2kπ, k∈Z" using the stated assumptions.
Before you start
Unit circle and periodicity.
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 2sin x=1 for all real x.
Why this math matters
Combine reference-angle solutions with full-period shifts. 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 unknown is a real angle in radians.
- The equation is solved over all real inputs.
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.
List every solution of a periodic equation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve 2sin x=1 for all real x.
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
sin x=1/2
Isolate the trigonometric function first.
Work through the mathematics
In [0,2π), solutions are π/6 and 5π/6
Sine is positive in the first two quadrants.
Check the conclusion
x=π/6+2kπ or x=5π/6+2kπ, k∈Z
Adding every integer full turn supplies the complete real solution set.
The result
x=π/6+2kπ or x=5π/6+2kπ, k∈Z
Adding every integer full turn supplies the complete real solution set.
Common mistakes to catch
- A calculator's principal value is not the full periodic solution set.
- The two branches should not omit their common integer-period condition.
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
How many solutions lie in [0,4π)?
Show a hint
There are two in each full period.
Reveal answer and explanation
Four
Add one full turn to each first-period solution.
Practice 2
Does x=7π/6 solve the equation?
Show a hint
Check the sine sign.
Reveal answer and explanation
No
Its sine is −1/2, producing −1 on the left.
Take the idea with you
Distinguish one observed cycle from all possible phase solutions.
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: Recognize an inverse trigonometric function's chosen output range
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