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
- Reduce a composition to its principal angle rather than cancelling blindly.
- Justify the conclusion "arccos(−1/2)=2π/3" using the stated assumptions.
Before you start
Cosine and inverse-function restrictions.
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
Evaluate arccos(cos(4π/3)).
Why this math matters
Reduce a composition to its principal angle rather than cancelling blindly. 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:
- Inverse trigonometric functions use their standard principal real branches.
- Angles are measured in 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.
Recognize an inverse trigonometric function's chosen output range
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Evaluate arccos(cos(4π/3)).
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
cos(4π/3)=−1/2
The original angle lies in the third quadrant.
Work through the mathematics
arccos returns angles in [0,π]
Restricting cosine to this interval makes it one-to-one.
Check the conclusion
arccos(−1/2)=2π/3
This is the allowed principal angle with the same cosine, not the original 4π/3.
The result
arccos(−1/2)=2π/3
This is the allowed principal angle with the same cosine, not the original 4π/3.
Common mistakes to catch
- Inverse notation does not mean reciprocal.
- Cancellation works only on the domain where the original function was restricted.
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
Evaluate arcsin(sin(3π/4)).
Show a hint
arcsin outputs in [−π/2,π/2].
Reveal answer and explanation
π/4
Both angles have sine √2/2, but only π/4 is in the principal range.
Practice 2
When does arccos(cos x)=x hold?
Show a hint
Keep x in the inverse's branch interval.
Reveal answer and explanation
For x∈[0,π]
Outside it, the composition selects a different representative.
Take the idea with you
Interpret a calculator's recovered angle when the physical motion may have completed extra turns.
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: Separate horizontal shift from angular frequency
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