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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Apply the Pythagorean identity.
- Choose a sign from geometric information.
- Evaluate two double-angle expressions.
Before you start
Unit-circle signs, fractions, and squaring numbers.
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
An angle θ lies in quadrant II and sin θ = 3/5. Find cos θ, sin(2θ), and cos(2θ) without approximating θ.
Why this math matters
Identities connect values exactly, avoiding rounding from a calculator-derived angle. The quadrant supplies information that a squared equation cannot retain on its own.
Set up the model
A useful answer starts with clear assumptions:
- θ is strictly between 90° and 180°.
- The value 3/5 is exact.
- The double-angle formulas are applied to the same θ throughout.
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.
Use an identity without losing the quadrant
PausedQuestion: Start with the question. Paused.
Question
Start with the question
An angle θ lies in quadrant II and sin θ = 3/5. Find cos θ, sin(2θ), and cos(2θ) without approximating θ.
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.
Recover the squared value
cos²θ = 1 − 9/25 = 16/25
Sine squared plus cosine squared equals one. At this point, either sign of cosine would satisfy the squared equation.
Use the quadrant
cos θ = −4/5
The unit-circle point lies left of the vertical axis in quadrant II, so its horizontal coordinate is negative.
Double the angle exactly
sin2θ = 2(3/5)(−4/5) = −24/25; cos2θ = 16/25 − 9/25 = 7/25
The double-angle identities use products and squares, not a doubling of the original sine or cosine value.
The result
cos θ = −4/5, sin(2θ) = −24/25, and cos(2θ) = 7/25.
The new values satisfy (−24/25)² + (7/25)² = 1. Their signs place the doubled angle's terminal side in quadrant IV.
Common mistakes to catch
- Taking only the positive square root contradicts quadrant II.
- sin(2θ) is not generally 2sin θ.
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
If sin θ = 1/2, find cos(2θ).
Show a hint
Use 1 − 2sin²θ.
Reveal answer and explanation
1/2
1 − 2(1/4) = 1/2, independent of the sign of cos θ.
Practice 2
Simplify (1 − cos²x)/sin x when sin x ≠ 0.
Show a hint
Replace the numerator using the identity.
Reveal answer and explanation
sin x on the stated domain
sin²x divided by sin x equals sin x; the original expression remains undefined where sin x = 0.
Take the idea with you
Track domain restrictions and quadrant information alongside algebraic identity manipulations.
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: Locate a point from a baseline and two angles
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