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Trigonometric identities and transformations

Read the idea, work independently, then explain what changed.

高一必修第一册(A版).pdf · 5.5 · PDF 222 / printed page 215

Revisit first: Graphs and properties of trigonometric functions

TOPIC 01

Trigonometric identities and transformations

Build understanding of trigonometric identities and transformations through definitions, contrasting cases and justified applications.

What you will be able to explain

Use trigonometric identities and transformations with explicit angle units and domains.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Angle sums

Use paired sine and cosine products in the addition identities.

\sin(a+b)=\sin a\cos b+\cos a\sin b

Equivalent forms

Choose a double-angle or half-angle form matching the given data.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in trigonometric identities and transformations changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

θ+φ=45° = 0.7854 rad; sin=0.7071, cos=0.7071. The circle has radius 1.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Given an acute angle, find sin 2α.

\sin\alpha=3/5

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Select an identity that matches the structure, retain signs and check denominators.

Hint 2

First determine the positive cosine.

Worked solution

Select an identity that matches the structure, retain signs and check denominators.
Calculate or simplify this relation.
$\cos\alpha=4/5;\quad\sin2\alpha=2(3/5)(4/5)=24/25$
The acute-angle condition selects the sign.

The requested value is 0.96.

Checks and common pitfalls: The acute-angle condition selects the sign.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Find cos 2α using the given sine.

\sin\alpha=5/13

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Select an identity that matches the structure, retain signs and check denominators.

Hint 2

Use the double-angle form containing only sine.

Worked solution

Select an identity that matches the structure, retain signs and check denominators.
Calculate or simplify this relation.
$\cos2\alpha=1-2\sin^2\alpha=1-50/169=119/169$
No sign of cosine is needed for this identity.

The requested value is 0.70414201.

Checks and common pitfalls: No sign of cosine is needed for this identity.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Prove the identity and state its domain restriction.

\frac{1-\cos2x}{\sin2x}=\tan x

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Select an identity that matches the structure, retain signs and check denominators.

Hint 2

Apply both double-angle identities.

Worked solution

Select an identity that matches the structure, retain signs and check denominators.
Calculate or simplify this relation.
$\frac{2\sin^2x}{2\sin x\cos x}=\frac{\sin x}{\cos x}$
Cancellation must preserve the exclusions of the original fraction.

It holds where sin 2x≠0.

Checks and common pitfalls: Cancellation must preserve the exclusions of the original fraction.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

For an acute α with sin α=5/13, find sin 2α.

\sin\alpha=5/13

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Determine cosine using the quadrant.

Hint 2

Use the 5–12–13 ratio.

Worked solution

Determine cosine using the quadrant.
Calculate or simplify this relation.
$\cos\alpha=12/13;\quad\sin2\alpha=2(5/13)(12/13)=120/169$
Both sine and cosine are positive for an acute angle.

The requested value is 0.71005917.

Checks and common pitfalls: Both sine and cosine are positive for an acute angle.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Given cos α=3/5, find cos 2α.

\cos\alpha=3/5

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Choose the double-angle form involving cosine alone.

Hint 2

Square the given cosine before doubling.

Worked solution

Choose the double-angle form involving cosine alone.
Calculate or simplify this relation.
$\cos2\alpha=2\cos^2\alpha-1=18/25-1=-7/25$
The answer need not have the same sign as cos α.

The requested value is -0.28.

Checks and common pitfalls: The answer need not have the same sign as cos α.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Evaluate sin 75° exactly.

\sin(45^{\circ}+30^{\circ})

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Select an identity that matches the structure, retain signs and check denominators.

Hint 2

Split 75° into two standard angles.

Worked solution

Select an identity that matches the structure, retain signs and check denominators.
Calculate or simplify this relation.
$\sin75^{\circ}=\frac{\sqrt2}2\frac{\sqrt3}2+\frac{\sqrt2}2\frac12=\frac{\sqrt6+\sqrt2}4$
Sine of a sum is not the sum of the sines.

(√6+√2)/4.

Checks and common pitfalls: Sine of a sum is not the sum of the sines.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Find tan(α+β), assuming both tangents have the stated values.

\tan\alpha=1,\quad\tan\beta=3

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Select an identity that matches the structure, retain signs and check denominators.

Hint 2

Use numerator sum and denominator one minus product.

Worked solution

Select an identity that matches the structure, retain signs and check denominators.
Calculate or simplify this relation.
$\tan(\alpha+\beta)=\frac{1+3}{1-3}$
The denominator 1−3 is nonzero.

The requested value is -2.

Checks and common pitfalls: The denominator 1−3 is nonzero.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

For π<α<3π/2 and cos α=−3/5, determine the sign and value of sin(α/2).

\pi<\alpha<3\pi/2,\quad\cos\alpha=-3/5;\quad\sin^2(\alpha/2)=(1-\cos\alpha)/2

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Select an identity that matches the structure, retain signs and check denominators.

Hint 2

Locate the half-angle before choosing a square-root sign.

Worked solution

Select an identity that matches the structure, retain signs and check denominators.
Calculate or simplify this relation.
$\pi/2<\alpha/2<3\pi/4;\quad\sin(\alpha/2)=\sqrt{4/5}$
The interval, not merely the terminal-ray quadrant, fixes the half-angle sign; adding 2π to α reverses that sign.

Positive, equal to 2/√5.

Checks and common pitfalls: The interval, not merely the terminal-ray quadrant, fixes the half-angle sign; adding 2π to α reverses that sign.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Find the amplitude of the combined sinusoid.

y=9\sin x+12\cos x

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Select an identity that matches the structure, retain signs and check denominators.

Hint 2

Represent the coefficients as perpendicular components.

Worked solution

Select an identity that matches the structure, retain signs and check denominators.
Calculate or simplify this relation.
$A=\sqrt{(9)^2+(12)^2}=15$
Amplitude is the Euclidean length of the coefficient pair.

The requested value is 15.

Checks and common pitfalls: Amplitude is the Euclidean length of the coefficient pair.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Prove the identity and preserve its original excluded inputs.

\frac{\sin^2x}{1-\cos x}=1+\cos x

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Factor a difference of squares.

Hint 2

Cancel only where 1−cos x is nonzero.

Worked solution

Factor a difference of squares.
Calculate or simplify this relation.
$\sin^2x=1-\cos^2x=(1-\cos x)(1+\cos x)$
The simplified right side exists at more inputs than the original left side.

Valid where cos x≠1.

Checks and common pitfalls: The simplified right side exists at more inputs than the original left side.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

For acute α and β, find sin(α−β) from the given values.

\sin\alpha=3/5,\quad\cos\alpha=4/5,\quad\sin\beta=5/13,\quad\cos\beta=12/13

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Match the sine-cosine pairs in the subtraction formula.

Hint 2

The two products are 36/65 and 20/65.

Worked solution

Match the sine-cosine pairs in the subtraction formula.
Calculate or simplify this relation.
$\sin(\alpha-\beta)=3/5\cdot12/13-4/5\cdot5/13=16/65$
The minus sign belongs between the products.

The requested value is 0.24615385.

Checks and common pitfalls: The minus sign belongs between the products.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Given tan α=2, find tan 2α.

\tan\alpha=2

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the tangent double-angle identity.

Hint 2

Check that 1−tan²α is nonzero.

Worked solution

Use the tangent double-angle identity.
Calculate or simplify this relation.
$\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}=4/(1-4)=-4/3$
The denominator is negative, determining the sign.

The requested value is -1.33333333.

Checks and common pitfalls: The denominator is negative, determining the sign.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Derive the product-to-sum identity for cos x cos y.

\cos x\cos y

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Add the two cosine addition formulas.

Hint 2

The sine-product terms cancel.

Worked solution

Add the two cosine addition formulas.
Calculate or simplify this relation.
$\cos(x-y)+\cos(x+y)=2\cos x\cos y$
No division by a trigonometric expression is needed, so the identity holds for all real x,y.

[cos(x−y)+cos(x+y)]/2.

Checks and common pitfalls: No division by a trigonometric expression is needed, so the identity holds for all real x,y.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

Choose a foundation skill to revisit ↗

Teacher preparation and assessment

Question sequence

What must be true before using the main rule for trigonometric identities and transformations?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Board plan

Use trigonometric identities and transformations with explicit angle units and domains.
Angle sums: Use paired sine and cosine products in the addition identities.
$\sin(a+b)=\sin a\cos b+\cos a\sin b$
Equivalent forms: Choose a double-angle or half-angle form matching the given data.
Close with: conditions → representation → reasoning → check.

Anticipated thinking

Expected reasoning: Use paired sine and cosine products in the addition identities.
Expected reasoning: Choose a double-angle or half-angle form matching the given data.
Expected correction: Cancelled factors cannot restore originally forbidden inputs.

Assessment checklist

1: identify the givens and required quantity.
1: choose a valid definition, representation or method.
1: present connected, correct reasoning.
1: check conditions and explain the result.