Building the cosine graph from circular motion

Textbook reading

The problem that opens the lesson

A rotating point begins at $(1,0)$ . At what inputs does its horizontal coordinate equal its vertical coordinate during one revolution?

Solution

Begin by identifying the mathematical object and the information that fixes it. Construct cosine from unit-circle key points, identify its extrema and zeros, and compare its graph to sine by horizontal translation. The relevant conditions are not optional bookkeeping: Equivalent phase-shift descriptions are not unique; adding whole periods produces the same function. Following that structure gives $t=\frac{pi}{4}$ and $\frac{5pi}{4}$ .

Why this works

Comparing sine and cosine on the same axes reveals that they describe perpendicular projections of one rotating point rather than unrelated formulas. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

The cosine graph records the horizontal coordinate of unit-circle motion.

Cosine begins at $1,$ moves through $0,-1,0,1,$ and repeats every $2pi$ . It is even because the points reached at $t$ and -t have the same horizontal coordinate. Cosine is a phase-shifted sine: cos $t=sin(t+\frac{pi}{2})$ .

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.

Textbook reading

Why the relationship works

Comparing sine and cosine on the same axes reveals that they describe perpendicular projections of one rotating point rather than unrelated formulas.

Textbook reading

A reliable way to work

Construct cosine from unit-circle key points, identify its extrema and zeros, and compare its graph to sine by horizontal translation.

Equivalent phase-shift descriptions are not unique; adding whole periods produces the same function.

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.

Textbook reading

What commonly goes wrong

A common error is to assume cosine has a different period or to shift in the wrong direction when rewriting it as sine.

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 begins at $(1,0)$ . At what inputs does its horizontal coordinate equal its vertical coordinate during one revolution?

Solution

Begin by identifying the mathematical object and the information that fixes it. Construct cosine from unit-circle key points, identify its extrema and zeros, and compare its graph to sine by horizontal translation. The relevant conditions are not optional bookkeeping: Equivalent phase-shift descriptions are not unique; adding whole periods produces the same function. Following that structure gives $t=\frac{pi}{4}$ and $\frac{5pi}{4}$ .

Why this works

Comparing sine and cosine on the same axes reveals that they describe perpendicular projections of one rotating point rather than unrelated formulas. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Build the cosine graph from quarter-cycle points.

Worked development

Construct cosine from unit-circle key points, identify its extrema and zeros, and compare its graph to sine by horizontal translation. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Cosine begins at $1,$ moves through $0,-1,0,1,$ and repeats every $2pi$ . It is even because the points reached at $t$ and -t have the same horizontal coordinate. Cosine is a phase-shifted sine: cos $t=sin(t+\frac{pi}{2})$ . Then apply the conditions explicitly: Equivalent phase-shift descriptions are not unique; adding whole periods produces the same function. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Cosine naturally models quantities that begin at a maximum or minimum.

Reasoning example

Problem

Show cos $t=sin(t+\frac{pi}{2})$ .

Worked development

Construct cosine from unit-circle key points, identify its extrema and zeros, and compare its graph to sine by horizontal translation. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Cosine begins at $1,$ moves through $0,-1,0,1,$ and repeats every $2pi$ . It is even because the points reached at $t$ and -t have the same horizontal coordinate. Cosine is a phase-shifted sine: cos $t=sin(t+\frac{pi}{2})$ . Then apply the conditions explicitly: Equivalent phase-shift descriptions are not unique; adding whole periods produces the same function. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Cosine naturally models quantities that begin at a maximum or minimum.

Worked example 4: quick check

Write cosine as a shifted sine function.

Solution

Begin by identifying the mathematical object and the information that fixes it. Construct cosine from unit-circle key points, identify its extrema and zeros, and compare its graph to sine by horizontal translation. The relevant conditions are not optional bookkeeping: Equivalent phase-shift descriptions are not unique; adding whole periods produces the same function. Following that structure gives cos $t=sin(t+\frac{pi}{2})$ .

Why this works

Comparing sine and cosine on the same axes reveals that they describe perpendicular projections of one rotating point rather than unrelated formulas. 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

Cosine naturally models quantities that begin at a maximum or minimum.

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