**Trigonometric ratios in similar right triangles :**

When we find the ratio of two sides in a triangle, the ratio of the corresponding sides in a similar triangle will always be the same. As such, this means that the trigonometric ratios (sine, cosine and tangent) in similar right-angle triangles are always equal.

**Example 1 :**

Find the cosec ∠E.

**Solution :**

In order to find cosec ∠E first we have to find the length all sides of triangle GEF.

By observing the above two given triangles, we can prove that the given triangles are similar.

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.

In triangles GEF and BCD,

∠G = ∠C (90 degree)

∠E = ∠B

Therefore, by the AA Similarity Theorem, △DEF~ △GIH. This means ratios of corresponding side lengths are equal.

(EF/BD) = (GF/CD) = (GE/BC)

cosec ∠E = Hypotenuse side/ Opposite side ==> (EF/GF)

Since (EF/GF) = (BD/CD), we can apply BD/CD instead of (EF/GF)

cosec ∠E = 37/35

**Example 2 :**

Find the sec ∠D.

**Solution :**

By observing the above two given triangles, we can prove that the given triangles are similar.

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.

In triangles GHI and DFE,

∠I = ∠E (90 degree)

∠G = ∠D

Therefore, by the AA Similarity theorem,

△GHI ~ △DFE.This means ratios of corresponding side lengths are equal.

(GH/DF) = (IH/EF) = (IG/ED)

sec ∠D = Hypotenuse side/ Adjacent side ==> (DF/ED)

Since (IH/GI) = (DF/ED), we can apply (IH/GI) instead of (DF/ED).

sec ∠D = 17/8

**Example 3 :**

Find the tan ∠P.

**Solution :**

By observing the above two given triangles, we can prove that the given triangles are similar.

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.

In triangles UST and PRQ,

∠S = ∠Q (90 degree)

∠T = ∠P

Therefore, by the AA Similarity theorem,

△UST ~ △PRQ.This means ratios of corresponding side lengths are equal.

(UT/RP) = (US/RQ) = (TS/PQ)

tan ∠P = Opposite side / Adjacent side ==> (RQ/PQ)

Since (US/TS) = (RQ/PQ), we can apply (US/TS) instead of (RQ/PQ).

tan ∠P = 20/21

After having gone through the stuff given above, we hope that the students would have understood "Trigonometric ratios in similar right triangles".

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