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Cosine
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Geometry Hub / Logic & Similarity / Cosine
10.05 • Logic & Similarity

Cosine

Read cosine from the adjacent side and the hypotenuse of a right triangle, and connect that ratio to one chosen acute angle.

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≈ Cosine
Interactive diagram

Cosine Diagram

Move the triangle, keep the marked angle fixed, and compare the adjacent side with the hypotenuse.

Use the movable diagram to see what defines cosine, how the labels relate to the figure, and what stays true as the board changes.

Definition: Cosine is the ratio of adjacent side to hypotenuse in a right triangle.
Detailed definition

Understanding Cosine

In right-triangle trigonometry, cosine of an acute angle is the ratio of the adjacent side to the hypotenuse. The adjacent side is the leg touching the chosen angle that is not the hypotenuse.

Cosine stays the same for all similar right triangles with the same acute angle, which is why a ratio from one triangle can be used as a stable function value.

This page keeps the angle mark, the adjacent side, and the hypotenuse visible together so cosine is read from the triangle itself before it becomes symbolic notation.

Key facts

Important ideas to remember

  • Cosine is the ratio of adjacent side to hypotenuse in a right triangle.
  • Cosine is commonly remembered with CAH: adjacent over hypotenuse.
  • The adjacent side depends on the chosen angle, so it changes if the reference angle changes.
  • Cosine is closely related to sine through the same right-triangle structure, but it uses a different leg of the triangle.
Where it is used

Where cosine shows up

  • Use cosine when a right-triangle problem involves the adjacent side and the hypotenuse relative to a chosen angle.
  • Use it in geometry and trigonometry problems involving horizontal distance, components, or side-length solving.
  • Use cosine inside the SOH CAH TOA framework when selecting the correct ratio from given information.
Common mistakes

What to watch out for

  • Do not confuse the adjacent side with the opposite side; both depend on the marked angle.
  • Do not use the hypotenuse as the adjacent side even though it touches the angle.
  • Do not switch angles halfway through a problem and keep the same side labels.
Worked examples

Cosine examples

Use these worked examples to see the idea in a clean diagram first, then in the kind of reasoning students usually need for classwork, homework, or test practice.

Example 1

Example 1: Picking the correct sides for cosine

Start from the marked acute angle, then name the two sides needed for the ratio before writing anything symbolic.

  • Locate the chosen angle.
  • Identify the opposite, adjacent, and hypotenuse relative to that angle.
  • Use only the side names that belong to the selected trig ratio.

Result: The ratio is set up correctly because the side names come from the angle, not from fixed positions on the page.

Example 2

Example 2: Checking a cosine ratio on a scaled triangle

Change the triangle size and confirm that the ratio idea still makes sense when the shape stays similar.

  • Read the side labels again after scaling.
  • Write the same trig ratio with the new lengths.
  • Compare the result with the original triangle.

Result: The ratio stays meaningful because similar right triangles preserve the same angle-based relationships.

For

Why this page helps

This page helps because cosine is easiest to understand when students can point to the angle, then point to the adjacent side that belongs to that angle. The board makes that relationship much more concrete.

Do

What you can do here

  • Track the adjacent side and hypotenuse directly from the marked angle.
  • Resize the triangle and confirm that the cosine ratio remains stable under similarity.
  • Keep a graph-free triangle view that makes the CAH part of SOH CAH TOA easy to review.
Learning outcome

What this page helps you do

These takeaways are meant to help you recognize the idea faster, read diagrams more accurately, and use the topic with more confidence in real problems.

1

Cosine

Use cosine as an angle-based side ratio with stronger confidence.

2

Cosine

Separate adjacent-side logic from opposite-side logic more reliably.

3

Cosine

Choose the correct trig ratio faster in right-triangle problems.

10

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Sine is the ratio of opposite side to hypotenuse in a right triangle.

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