Sine and cosine as coordinate functions

Textbook reading

The problem that opens the lesson

A rotating point has unit-circle coordinate $P(t)=(-\frac{\frac{5}{13,12}}{13})$ . Find sin t, cos t, and the quadrant.

Solution

Begin by identifying the mathematical object and the information that fixes it. Translate among point coordinates, function values, and equations. To solve sin $t=k$ or cos $t=k$ on one revolution, find every unit-circle point with the required coordinate. The relevant conditions are not optional bookkeeping: Not every pair (cos t,sin t) can be chosen independently; the pair must lie on the unit circle. Following that structure gives sin $t=\frac{12}{13,}$ cos $t=-\frac{5}{13,}$ quadrant II.

Why this works

A known coordinate immediately gives sine and cosine. If one coordinate is known, use the unit-circle equation to find the other, then use the quadrant to choose the correct sign. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

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What this lesson is really about

If $P(t)=(x,y)$ is the unit-circle point associated with t, then cos $t=x$ and sin $t=y$ .

The definitions make sine and cosine functions of every real input. The identity $x^2+y^2=1$ becomes $sin^2 t+cos^2 t=1$ . Coordinate signs give quadrant signs, and the coordinate bounds produce ranges $[-1,1]$ .

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

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Why the relationship works

A known coordinate immediately gives sine and cosine. If one coordinate is known, use the unit-circle equation to find the other, then use the quadrant to choose the correct sign.

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A reliable way to work

Translate among point coordinates, function values, and equations. To solve sin $t=k$ or cos $t=k$ on one revolution, find every unit-circle point with the required coordinate.

Not every pair (cos t,sin t) can be chosen independently; the pair must lie on the unit circle.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

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What commonly goes wrong

A common error is to reverse sine and cosine coordinates or to choose both coordinate signs without using quadrant information.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

A rotating point has unit-circle coordinate $P(t)=(-\frac{\frac{5}{13,12}}{13})$ . Find sin t, cos t, and the quadrant.

Solution

Begin by identifying the mathematical object and the information that fixes it. Translate among point coordinates, function values, and equations. To solve sin $t=k$ or cos $t=k$ on one revolution, find every unit-circle point with the required coordinate. The relevant conditions are not optional bookkeeping: Not every pair (cos t,sin t) can be chosen independently; the pair must lie on the unit circle. Following that structure gives sin $t=\frac{12}{13,}$ cos $t=-\frac{5}{13,}$ quadrant II.

Why this works

A known coordinate immediately gives sine and cosine. If one coordinate is known, use the unit-circle equation to find the other, then use the quadrant to choose the correct sign. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Use $x^2+y^2=1$ to recover a missing coordinate.

Worked development

Translate among point coordinates, function values, and equations. To solve sin $t=k$ or cos $t=k$ on one revolution, find every unit-circle point with the required coordinate. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. The definitions make sine and cosine functions of every real input. The identity $x^2+y^2=1$ becomes $sin^2 t+cos^2 t=1$ . Coordinate signs give quadrant signs, and the coordinate bounds produce ranges $[-1,1]$ . Then apply the conditions explicitly: Not every pair (cos t,sin t) can be chosen independently; the pair must lie on the unit circle. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Coordinate functions model circular position, oscillation, projection, sound, waves, and periodic change.

Reasoning example

Problem

Determine the signs of sine and cosine in each quadrant.

Worked development

Translate among point coordinates, function values, and equations. To solve sin $t=k$ or cos $t=k$ on one revolution, find every unit-circle point with the required coordinate. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. The definitions make sine and cosine functions of every real input. The identity $x^2+y^2=1$ becomes $sin^2 t+cos^2 t=1$ . Coordinate signs give quadrant signs, and the coordinate bounds produce ranges $[-1,1]$ . Then apply the conditions explicitly: Not every pair (cos t,sin t) can be chosen independently; the pair must lie on the unit circle. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Coordinate functions model circular position, oscillation, projection, sound, waves, and periodic change.

Worked example 4: quick check

If sin $t=-\frac{4}{5}$ and cos $t>0,$ find cos $t$ and the quadrant.

Solution

Begin by identifying the mathematical object and the information that fixes it. Translate among point coordinates, function values, and equations. To solve sin $t=k$ or cos $t=k$ on one revolution, find every unit-circle point with the required coordinate. The relevant conditions are not optional bookkeeping: Not every pair (cos t,sin t) can be chosen independently; the pair must lie on the unit circle. Following that structure gives cos $t=\frac{3}{5,}$ quadrant IV.

Why this works

A known coordinate immediately gives sine and cosine. If one coordinate is known, use the unit-circle equation to find the other, then use the quadrant to choose the correct sign. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

Application and interpretation

Coordinate functions model circular position, oscillation, projection, sound, waves, and periodic change.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Practice