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Trigonometric functions: mixed assessment

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

TOPIC 01

Trigonometric functions: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

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

01 / Foundation#Your turn

Write the angle as cπ radians; find c.

210^{\circ}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Keep degrees and radians distinct and use the definition of angular measure.

Hint 2

One straight angle is π radians.

Worked solution

Keep degrees and radians distinct and use the definition of angular measure.
Calculate or simplify this relation.
$210^{\circ}\cdot\frac{\pi}{180^{\circ}}=\frac{7}6\pi$
The numerical coefficient changes with the unit, but the geometric angle does not.

The requested value is 1.16666667.

Checks and common pitfalls: The numerical coefficient changes with the unit, but the geometric angle does not.

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

02 / Foundation#Your turn

Solve both parts and justify the conditions used. Part A: Find cos θ when θ is in quadrant II. Part B: Two daily temperature observations fit a sinusoid. Explain why more observations are needed before predicting its period.

\begin{gathered}\text{A: }\sin\theta=3/5\end{gathered}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use coordinates on the terminal ray and a positive radius.

Hint 2

B: Define the geometric or periodic quantities with units, then choose a trigonometric relation.

Worked solution

Part A reasoning
Use coordinates on the terminal ray and a positive radius.
Calculate or simplify this relation.
$\cos^2\theta=1-9/25=16/25;\quad\cos\theta=-4/5$
The quadrant determines the negative cosine.
Part B reasoning
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Record enough observations over several cycles and compare a fitted curve with residual errors and context.
A visual fit alone does not establish a unique periodic model.

A: The requested value is -0.8. B: Several amplitudes, phases and periods can fit two observations.

Checks and common pitfalls: The quadrant determines the negative cosine. A visual fit alone does not establish a unique periodic model.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

03 / Foundation#Your turn

Find tangent after a half turn.

\tan\alpha=7;\quad\tan(\pi+\alpha)

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use unit-circle symmetry and periodicity; determine the sign before simplifying.

Hint 2

Both sine and cosine change sign.

Worked solution

Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Calculate or simplify this relation.
$\tan(\pi+\alpha)=\tan\alpha=7$
Their quotient stays unchanged when defined.

The requested value is 7.

Checks and common pitfalls: Their quotient stays unchanged when defined.

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

04 / Foundation#Your turn

Count the zeros on the closed interval.

y=\sin x;\quad0\le x\le7\pi

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Read amplitude, period and allowed intervals directly from the trigonometric graph.

Hint 2

Include both interval endpoints if they are zeros.

Worked solution

Read amplitude, period and allowed intervals directly from the trigonometric graph.
Calculate or simplify this relation.
$x=n\pi,\quad n=0,1,\ldots,7$
The number of subintervals differs from the number of endpoints.

The requested value is 8.

Checks and common pitfalls: The number of subintervals differs from the number of endpoints.

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

05 / Foundation#Your turn

Solve both parts and justify the conditions used. Part A: For π<α<3π/2 and cos α=−3/5, determine the sign and value of sin(α/2). Part B: Evaluate without a calculator.

\begin{gathered}\text{A: }\pi<\alpha<3\pi/2,\quad\cos\alpha=-3/5;\quad\sin^2(\alpha/2)=(1-\cos\alpha)/2\\\text{B: }\sin(\pi)+\sin(0)\end{gathered}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

B: Use unit-circle symmetry and periodicity; determine the sign before simplifying.

Worked solution

Part A reasoning
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.
Part B reasoning
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Calculate or simplify this relation.
$\sin(\pi)+\sin(0)=0$
The sine coordinate is zero at integer multiples of π.

A: Positive, equal to 2/√5. B: The requested value is 0.

Checks and common pitfalls: The interval, not merely the terminal-ray quadrant, fixes the half-angle sign; adding 2π to α reverses that sign. The sine coordinate is zero at integer multiples of π.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

06 / Foundation#Your turn

What function remains when the frequency is zero? Does it have a least positive period?

y=7\sin(0x+\pi/2)

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Separate amplitude, internal frequency, phase and vertical shift.

Hint 2

Every positive shift preserves a constant function.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$y=7;\quad f(x+T)=f(x)\ (T>0)$
Because arbitrarily small positive shifts work, there is no least one.

The constant 7; it has no least positive period.

Checks and common pitfalls: Because arbitrarily small positive shifts work, there is no least one.

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 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: Two daily temperature observations fit a sinusoid. Explain why more observations are needed before predicting its period. Part B: Given an acute angle, find sin 2α.

\begin{gathered}\text{B: }\sin\alpha=3/5\end{gathered}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Define the geometric or periodic quantities with units, then choose a trigonometric relation.

Hint 2

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

Worked solution

Part A reasoning
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Record enough observations over several cycles and compare a fitted curve with residual errors and context.
A visual fit alone does not establish a unique periodic model.
Part B reasoning
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.

A: Several amplitudes, phases and periods can fit two observations. B: The requested value is 0.96.

Checks and common pitfalls: A visual fit alone does not establish a unique periodic model. The acute-angle condition selects the sign.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

08 / Standard#Your turn

Find the sector area for the stated radian angle.

r=8,\quad\theta=1

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Keep degrees and radians distinct and use the definition of angular measure.

Hint 2

A sector takes θ/(2π) of the circle area.

Worked solution

Keep degrees and radians distinct and use the definition of angular measure.
Calculate or simplify this relation.
$S=\frac12r^2\theta=\frac{64}2$
The area unit is the square of the radius unit.

The requested value is 32.

Checks and common pitfalls: The area unit is the square of the radius unit.

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

09 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: Determine whether the two claimed trigonometric values can occur together. Part B: A wheel has radius 8 m and its centre is 10 m above ground. Find the greatest passenger height.

\begin{gathered}\text{A: }\sin\theta=4/5,\quad\cos\theta=4/5\end{gathered}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use coordinates on the terminal ray and a positive radius.

Hint 2

B: Define the geometric or periodic quantities with units, then choose a trigonometric relation.

Worked solution

Part A reasoning
Use coordinates on the terminal ray and a positive radius.
Calculate or simplify this relation.
$(4/5)^2+(4/5)^2=32/25\ne1$
Each value being individually between −1 and 1 is insufficient.
Part B reasoning
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Calculate or simplify this relation.
$h_{\max}=(10)+8=18$
The corresponding minimum is 2 m, so the wheel remains above ground.

A: No, because their squared sum is not one. B: The requested value is 18.

Checks and common pitfalls: Each value being individually between −1 and 1 is insufficient. The corresponding minimum is 2 m, so the wheel remains above ground.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

10 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: Evaluate without a calculator. Part B: How many coterminal representatives of 30° lie in this closed interval?

\begin{gathered}\text{A: }\sin(\pi)+\sin(0)\\\text{B: }[-3240^{\circ},3240^{\circ}]\end{gathered}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use unit-circle symmetry and periodicity; determine the sign before simplifying.

Hint 2

B: Keep degrees and radians distinct and use the definition of angular measure.

Worked solution

Part A reasoning
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Calculate or simplify this relation.
$\sin(\pi)+\sin(0)=0$
The sine coordinate is zero at integer multiples of π.
Part B reasoning
Keep degrees and radians distinct and use the definition of angular measure.
Calculate or simplify this relation.
$-3240\le30+360n\le3240\Rightarrow -9\le n\le8$
Integer n must satisfy both bounds.

A: The requested value is 0. B: The requested value is 18.

Checks and common pitfalls: The sine coordinate is zero at integer multiples of π. Integer n must satisfy both bounds.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

11 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: Is sine increasing on the whole real line? Give a counterexample. Part B: Write the angle as cπ radians; find c.

\begin{gathered}\text{A: }y=\sin x\\\text{B: }210^{\circ}\end{gathered}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Read amplitude, period and allowed intervals directly from the trigonometric graph.

Hint 2

B: Keep degrees and radians distinct and use the definition of angular measure.

Worked solution

Part A reasoning
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Calculate or simplify this relation.
$\sin(\pi/2)=1>0=\sin\pi$
Monotonicity must be specified on suitable intervals.
Part B reasoning
Keep degrees and radians distinct and use the definition of angular measure.
Calculate or simplify this relation.
$210^{\circ}\cdot\frac{\pi}{180^{\circ}}=\frac{7}6\pi$
The numerical coefficient changes with the unit, but the geometric angle does not.

A: No: π/2<π but sin(π/2)>sin π. B: The requested value is 1.16666667.

Checks and common pitfalls: Monotonicity must be specified on suitable intervals. The numerical coefficient changes with the unit, but the geometric angle does not.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

12 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: Given an acute angle, find sin 2α. Part B: Find cos θ when θ is in quadrant II.

\begin{gathered}\text{A: }\sin\alpha=3/5\\\text{B: }\sin\theta=3/5\end{gathered}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

B: Use coordinates on the terminal ray and a positive radius.

Worked solution

Part A reasoning
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.
Part B reasoning
Use coordinates on the terminal ray and a positive radius.
Calculate or simplify this relation.
$\cos^2\theta=1-9/25=16/25;\quad\cos\theta=-4/5$
The quadrant determines the negative cosine.

A: The requested value is 0.96. B: The requested value is -0.8.

Checks and common pitfalls: The acute-angle condition selects the sign. The quadrant determines the negative cosine.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

13 / Transfer#Your turn

Write the least positive period as cπ; find c.

y=3\sin(8x+\pi/4)

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Separate amplitude, internal frequency, phase and vertical shift.

Hint 2

Phase does not change the cycle length.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$T=2\pi/8$
The nonzero amplitude and frequency guarantee a nonconstant sinusoid.

The requested value is 0.25.

Checks and common pitfalls: The nonzero amplitude and frequency guarantee a nonconstant sinusoid.

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

14 / Transfer#Your turn

A wheel has radius 8 m and its centre is 10 m above ground. Find the greatest passenger height.

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Define the geometric or periodic quantities with units, then choose a trigonometric relation.

Hint 2

At the top, add radius to centre height.

Worked solution

Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Calculate or simplify this relation.
$h_{\max}=(10)+8=18$
The corresponding minimum is 2 m, so the wheel remains above ground.

The requested value is 18.

Checks and common pitfalls: The corresponding minimum is 2 m, so the wheel remains above ground.

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

15 / Transfer#Your turn

How many coterminal representatives of 30° lie in this closed interval?

[-3240^{\circ},3240^{\circ}]

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Keep degrees and radians distinct and use the definition of angular measure.

Hint 2

Write every coterminal angle as 30°+360°n.

Worked solution

Keep degrees and radians distinct and use the definition of angular measure.
Calculate or simplify this relation.
$-3240\le30+360n\le3240\Rightarrow -9\le n\le8$
Integer n must satisfy both bounds.

The requested value is 18.

Checks and common pitfalls: Integer n must satisfy both bounds.

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

End-of-lesson check and correction

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Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.