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
- Find a doubled angle's sine and cosine without guessing its quadrant.
- Justify the conclusion "cos2θ=cos²θ−sin²θ=−7/25" using the stated assumptions.
Before you start
Unit-circle signs and Pythagorean identity.
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
If cos θ=3/5 and θ is in quadrant IV, find sin2θ and cos2θ.
Why this math matters
Find a doubled angle's sine and cosine without guessing its quadrant. 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:
- θ lies in quadrant IV.
- The cosine value is exact.
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.
Keep quadrant information in a double-angle calculation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
If cos θ=3/5 and θ is in quadrant IV, find sin2θ and cos2θ.
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 θ=−4/5
The magnitude follows from sin²+cos²=1, while the quadrant determines the negative sign.
Work through the mathematics
sin2θ=2sinθcosθ=−24/25
The double-angle sine uses both signed components.
Check the conclusion
cos2θ=cos²θ−sin²θ=−7/25
Both results place the doubled direction in quadrant III modulo a full turn.
The result
cos2θ=cos²θ−sin²θ=−7/25
Both results place the doubled direction in quadrant III modulo a full turn.
Common mistakes to catch
- Taking a square root requires the correct sign choice.
- The quadrant of θ does not automatically equal the quadrant of 2θ.
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
Find tan2θ.
Show a hint
Divide the doubled sine by cosine.
Reveal answer and explanation
24/7
The ratio of two negative values is positive.
Practice 2
Would choosing sinθ=+4/5 preserve the premise?
Show a hint
Check the stated quadrant.
Reveal answer and explanation
No
That would describe quadrant I, not quadrant IV.
Take the idea with you
Use signed coordinates to track a doubled rotational phase.
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: State where a simplified identity is actually defined
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