IXL - Trigonometric ratios in similar right triangles (Geometry practice)

Find the cosine of

∠

Simplify your answer and write it as a proper fraction, improper fraction, or whole number.

cos

(

)

ref_doc_title.

Learn with an example

question

Find the cosine of

∠

Simplify your answer and write it as a proper fraction, improper fraction, or whole number.

cos

(

)

key idea

Two triangles are similar if and only if they are scaled versions of each other. The ratios of their corresponding side lengths are equal and the measures of their corresponding angles are equal.

The AA Similarity Theorem states that two triangles are similar if two angles of one triangle are congruent to two angles of the other triangle.

△

∠

≅

∠

and

∠

≅

∠

The cosine

(

cos

)

of an angle in a right triangle is a ratio. It is the length of the adjacent leg

(

adj

)

divided by the length of the hypotenuse

(

hyp

)

The adjacent leg is the leg next to the specified angle and the hypotenuse is the side across from the right angle.

cos

(

)

adj

hyp

solution

By the definition of cosine,

cos

(

)

You don't know any side lengths in

△

but you do know the side lengths of

△

If you can prove a similarity between the two triangles, then you can find

using the side lengths of

△

∠

≅

∠

since angles with matching arcs are congruent.

∠

≅

∠

since all right angles are congruent. Therefore, by the AA Similarity Theorem,

△

This means ratios of corresponding side lengths are equal.

cos

(

)

Definition of cosine

Substitute

by similarity

Substitute

by similarity

So,

cos

(

)

Looking for more? Try these:

Lesson: Trigonometric ratios

Lesson: Similar triangles

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Teacher tools

Work it out

Not feeling ready yet? These can help:

M.8Similarity rules for triangles

Q.2Trigonometric ratios with radicals: sin, cos, and tan

Lesson: Trigonometric ratios

Lesson: Similar triangles

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