Trignometric Ratios & Pythagorean for Side Lengths (Mixed Practice)

Use the cosine ratio:

cos 30° = adjacent / hypotenuse

So, first write the equation:

cos 30° = 5 / H

cos 30° = √3 / 2, so:

√3 / 2 = 5 / H

Now solve for H:

H = 5 ÷ (√3 / 2) = 5 × (2 / √3) = 10 / √3 = (10√3) / 3

Hence, the hypotenuse = 10√3 / 3.

We are given sin α = 0.5 and cos α = √3 / 2.

Use the definition of tangent:

tan α = sin α / cos α

So:

tan α = (1/2) / (√3/2) = 1 / √3

Hence, tan α = 1/√3.

We are given tan θ = 5/12 and hypotenuse = 13.

The triple 5–12–13 is a standard right triangle:

opposite : adjacent : hypotenuse = 5 : 12 : 13

So if the hypotenuse is 13, the adjacent side is 12.

Hence, the adjacent side = 12.

We are given cos θ = 0.6.

First find sin θ using the identity:

sin θ = √(1 − cos²θ)

= √(1 − 0.6²)

= √(1 − 0.36)

= √0.64 = 0.8

Now find tan θ:

tan θ = sin θ / cos θ

= 0.8 / 0.6 = 8/6 = 4/3 ≈ 1.33

Hence, tan θ ≈ 1.33 (which is 4/3).

We are told: adjacent = 9, opposite = 12.

First find the hypotenuse:

hypotenuse = √(adjacent² + opposite²)

= √(9² + 12²)

= √(81 + 144) = √225 = 15

Now,

sin θ = opposite / hypotenuse = 12 / 15

Hence, hyp = 15 and sin θ = 12/15.

We are given tan θ = 3/4 and hypotenuse = 25.

The basic 3–4–5 triangle has sides:

opposite : adjacent : hypotenuse = 3 : 4 : 5

Scale by k so that 5k = 25 ⇒ k = 5:

opposite = 3k = 15

adjacent = 4k = 20

Hence, opposite = 15 and adjacent = 20.

We are given cos θ = 0.8 and hypotenuse = 25.

First find the adjacent side using cosine:

adjacent = hypotenuse × cos θ

adjacent = 25 × 0.8 = 20

Now use Pythagoras to find the opposite side:

opposite² = hypotenuse² − adjacent²

= 25² − 20²

= 625 − 400 = 225

So opposite = √225 = 15

Hence, the opposite side = 15.

We are given sin x = 5/13 and hypotenuse = 26.

First find cos x using the identity:

cos x = √(1 − sin²x)

= √(1 − (5/13)²)

= √(1 − 25/169)

= √(144/169) = 12/13

Now find the adjacent side:

adjacent = hypotenuse × cos x

adjacent = 26 × (12/13) = 24

Hence, cos x = 12/13 and adjacent = 24.

We are given sin θ = 24/25.

First find cos θ using the identity:

cos θ = √(1 − sin²θ)

= √(1 − (24/25)²)

= √(1 − 576/625)

= √(49/625) = 7/25

Now find tan θ:

tan θ = sin θ / cos θ

= (24/25) / (7/25) = 24/7

Hence, cos θ = 7/25 and tan θ = 24/7.

We are given sin B = 3/5 and the hypotenuse = 20.

First find cos B using the Pythagorean identity:

cos B = √(1 − sin²B)

= √(1 − (3/5)²)

= √(1 − 9/25)

= √(16/25) = 4/5

Now find the adjacent side:

adjacent = hypotenuse × cos B

adjacent = 20 × (4/5) = 16

Hence, the adjacent side = 16.

We are given tan θ = 4/3 and the opposite side = 8.

Use the tangent ratio:

tan θ = opposite / adjacent = 4/3

So:

4/3 = 8 / adjacent

Solve for adjacent:

adjacent = 8 ÷ (4/3) = 8 × 3/4 = 6

Now find the hypotenuse with Pythagoras:

hyp² = opposite² + adjacent²

= 8² + 6²

= 64 + 36 = 100

So hypotenuse = √100 = 10

Hence, adjacent = 6 and hypotenuse = 10.

We are given opposite = 9, adjacent = 40.

First find the hypotenuse:

hypotenuse = √(9² + 40²)

= √(81 + 1600)

= √1681 = 41

Now find sin θ:

sin θ = opposite / hypotenuse

= 9 / 41

Hence, hypotenuse = 41 and sin θ = 9/41.

Use the tangent ratio:

opposite = adjacent × tan 35°

So:

opposite ≈ 12 × tan 35°

≈ 12 × 0.7002 ≈ 8.4

Hence, the opposite side ≈ 8.4.

We are given opposite = 10 and tan θ = 2.

Use the tangent ratio:

tan θ = opposite / adjacent = 2

So:

2 = 10 / adjacent

adjacent = 10 / 2 = 5

Now find the hypotenuse using Pythagoras:

hyp² = 10² + 5² = 100 + 25 = 125

hypotenuse = √125 ≈ 11.2

Hence, adjacent ≈ 5 and hypotenuse ≈ 11.2.

We are given angle θ = 70° and hypotenuse = 40.

Use the sine ratio:

opposite = hypotenuse × sin θ

So:

opposite = 40 × sin 70° ≈ 40 × 0.9397 ≈ 37.6

Hence, the opposite side ≈ 37.6.

We are given adjacent = 9 and θ = 40°.

First find the opposite side using tangent:

opposite = adjacent × tan 40°

≈ 9 × 0.8391 ≈ 7.6

Now find the hypotenuse using cosine:

hypotenuse = adjacent / cos 40°

≈ 9 / 0.7660 ≈ 11.8

Hence, opposite ≈ 7.6 and hypotenuse ≈ 11.8.

In a 45°–45°–90° triangle, both legs are equal and:

hypotenuse = leg × √2

Given leg = 7:

hypotenuse = 7 × √2 = 7√2

Hence, the hypotenuse = 7√2.

We are given tan β = 1.5 and adjacent = 6.

First find the opposite side using tangent:

tan β = opposite / adjacent

So:

1.5 = opposite / 6

opposite = 6 × 1.5 = 9.0

Now find the hypotenuse with Pythagoras:

hyp² = 6² + 9² = 36 + 81 = 117

hypotenuse = √117 ≈ 10.8

Hence, opposite ≈ 9.0 and hypotenuse ≈ 10.8.

Use the cosine ratio:

cos 60° = adjacent / hypotenuse

So: cos 60° = 9 / H

cos 60° = 0.5, so:

0.5 = 9 / H

Solve for H:

H = 9 / 0.5 = 18

Hence, the hypotenuse = 18.

We are given sin α = 4/5 and opposite = 24.

First find the hypotenuse using the sine ratio:

sin α = opposite / hypotenuse

So:

4/5 = 24 / hyp

hyp = 24 ÷ (4/5) = 24 × 5/4 = 30

Now find cos α using the identity:

cos α = √(1 − sin²α)

= √(1 − (4/5)²)

= √(1 − 16/25)

= √(9/25) = 3/5

Then adjacent = hyp × cos α

adjacent = 30 × (3/5) = 18

Hence, hypotenuse = 30 and adjacent = 18.