Mathspresso
Math II ยท G-SRT.8.1

Deriving and Using Trig Ratios for 30-60-90 and 45-45-90 Special Right Triangles

Special right triangles give exact trig values, helping students see trigonometry as geometry rather than decimal-button punching.

Concept Geometry
Domain Similarity, Right Triangles, and Trigonometry
Read time 1 minutes

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Problem Library

Problems in the App From This Objective

144 problems across 12 archetypes in the app.

use `x, x, x sqrt(2)`.
12 problems Warmup Practice Mixed Review Assessment
Problem 1

Identify side ratios in a 45-45-90 triangle when leg length is x.

Problem 2

Identify side ratios in a 45-45-90 triangle when hypotenuse is h.

Problem 3

Identify side ratios in a 45-45-90 triangle when one leg is 5.

Problem 4

Identify side ratios in a 45-45-90 triangle when hypotenuse is 10.

Problem 5

Identify side ratios in a 45-45-90 triangle when one leg is 7.

Open in simulator
Problem 6

Identify side ratios in a 45-45-90 triangle when hypotenuse is 8.

Problem 7

Identify side ratios in a 45-45-90 triangle when a leg has length 3 sqrt(2).

Problem 8

Identify side ratios in a 45-45-90 triangle when the hypotenuse measures 12 sqrt(2).

Problem 9

Identify side ratios in a 45-45-90 triangle when a leg is length a.

Problem 10

Identify side ratios in a 45-45-90 triangle when the hypotenuse is k.

Problem 11

Identify side ratios in a 45-45-90 triangle when one leg is 2x.

Problem 12

Identify side ratios in a 45-45-90 triangle when hypotenuse is 6y.

use `x, x sqrt(3), 2x`.
12 problems Warmup Practice Mixed Review Assessment
Problem 13

Identify side ratios in a 30-60-90 triangle when short leg is x.

Problem 14

Identify side ratios in a 30-60-90 triangle when hypotenuse is h.

Problem 15

Identify side ratios in a 30-60-90 triangle when long leg is L.

Problem 16

Identify side ratios in a 30-60-90 triangle when short leg is 4.

Problem 17

Identify side ratios in a 30-60-90 triangle when hypotenuse is 10.

Problem 18

Identify side ratios in a 30-60-90 triangle when long leg is 9.

Problem 19

Identify side ratios in a 30-60-90 triangle when short leg is 7.

Problem 20

Identify side ratios in a 30-60-90 triangle when hypotenuse is 12.

Open in simulator
Problem 21

Identify side ratios in a 30-60-90 triangle when long leg is 5sqrt(3).

Problem 22

Identify side ratios in a 30-60-90 triangle when short leg is 2sqrt(3).

Problem 23

Identify side ratios in a 30-60-90 triangle when hypotenuse is 8sqrt(3).

Problem 24

Identify side ratios in a 30-60-90 triangle when long leg is 6.

apply special triangle ratio.
12 problems Warmup Practice Mixed Review Assessment
Problem 25

Find missing sides in a 45-45-90 triangle from one leg is 9.

Problem 26

Find missing sides in a 45-45-90 triangle from hypotenuse is 14.

Open in simulator
Problem 27

Find missing sides in a 45-45-90 triangle from one leg is a.

Problem 28

Find missing sides in a 45-45-90 triangle from hypotenuse is h.

Problem 29

Find missing sides in a 45-45-90 triangle from one leg is 5.

Problem 30

Find missing sides in a 45-45-90 triangle from hypotenuse is 10.

Problem 31

Find missing sides in a 45-45-90 triangle from one leg is 3sqrt(2).

Problem 32

Find missing sides in a 45-45-90 triangle from hypotenuse is 8sqrt(2).

Problem 33

Find missing sides in a 45-45-90 triangle from one leg is 7.

Problem 34

Find missing sides in a 45-45-90 triangle from hypotenuse is 20.

Problem 35

Find missing sides in a 45-45-90 triangle from one leg is 6sqrt(3).

Problem 36

Find missing sides in a 45-45-90 triangle from hypotenuse is 12sqrt(2).

match side opposite angle to ratio.
12 problems Warmup Practice Mixed Review Assessment
Problem 37

Find missing sides in a 30-60-90 triangle from short leg is 6.

Problem 38

Find missing sides in a 30-60-90 triangle from hypotenuse is 18.

Problem 39

Find missing sides in a 30-60-90 triangle from long leg is 12sqrt(3).

Open in simulator
Problem 40

Find missing sides in a 30-60-90 triangle from long leg is L.

Problem 41

Find missing sides in a 30-60-90 triangle from short leg is 5.

Problem 42

Find missing sides in a 30-60-90 triangle from hypotenuse is 20.

Problem 43

Find missing sides in a 30-60-90 triangle from long leg is 9.

Problem 44

Find missing sides in a 30-60-90 triangle from short leg is 4sqrt(3).

Problem 45

Find missing sides in a 30-60-90 triangle from hypotenuse is 10sqrt(3).

Problem 46

Find missing sides in a 30-60-90 triangle from short leg is K.

Problem 47

Find missing sides in a 30-60-90 triangle from hypotenuse is H.

Problem 48

Find missing sides in a 30-60-90 triangle from long leg is M.

use Pythagorean Theorem.
12 problems Warmup Practice Mixed Review Assessment
Problem 49

Derive the 45-45-90 ratio from an isosceles right triangle with legs both x.

Problem 50

Derive the 45-45-90 ratio from an isosceles right triangle with legs both 1.

Problem 51

Derive the 45-45-90 ratio from an isosceles right triangle with legs both 5.

Problem 52

Derive the 45-45-90 ratio from an isosceles right triangle with equal legs a.

Problem 53

Derive the 45-45-90 ratio from an isosceles right triangle with legs both y.

Problem 54

Derive the 45-45-90 ratio from an isosceles right triangle with legs both 2.

Problem 55

Derive the 45-45-90 ratio from an isosceles right triangle with legs both 3.

Problem 56

Derive the 45-45-90 ratio from an isosceles right triangle with legs both 1/2.

Open in simulator
Problem 57

Derive the 45-45-90 ratio from an isosceles right triangle with legs both 2x.

Problem 58

Derive the 45-45-90 ratio from an isosceles right triangle with legs both 3k.

Problem 59

Derive the 45-45-90 ratio from an isosceles right triangle with equal sides s.

Problem 60

Derive the 45-45-90 ratio from an isosceles right triangle with legs both 4.

split equilateral triangle and use Pythagorean Theorem.
12 problems Warmup Practice Mixed Review Assessment
Problem 61

Derive the 30-60-90 ratio from an equilateral triangle with side length 2x.

Problem 62

Derive the 30-60-90 ratio from an equilateral triangle with side length 2.

Problem 63

Derive the 30-60-90 ratio from an equilateral triangle with side length s.

Problem 64

Derive the 30-60-90 ratio from an equilateral triangle with equilateral triangle side 10.

Problem 65

Derive the 30-60-90 ratio from an equilateral triangle with side length 4.

Problem 66

Derive the 30-60-90 ratio from an equilateral triangle with equilateral triangle side 6.

Problem 67

Derive the 30-60-90 ratio from an equilateral triangle with side length 8.

Problem 68

Derive the 30-60-90 ratio from an equilateral triangle with side length 2a.

Problem 69

Derive the 30-60-90 ratio from an equilateral triangle with side length 4y.

Problem 70

Derive the 30-60-90 ratio from an equilateral triangle with equilateral triangle side 12.

Problem 71

Derive the 30-60-90 ratio from an equilateral triangle with an equilateral triangle with side length 'm'.

Problem 72

Derive the 30-60-90 ratio from an equilateral triangle with side length 20.

Open in simulator
use special right triangle ratios.
12 problems Warmup Practice Mixed Review Assessment
Problem 73

Find exact sine, cosine, and tangent for 45 degrees using legs 1 and 1, hypotenuse sqrt(2).

Problem 74

Find exact sine, cosine, and tangent for 45 degrees using legs x and x, hypotenuse xsqrt(2).

Problem 75

Find exact sine, cosine, and tangent for 45 degrees using legs 5 and 5, hypotenuse 5sqrt(2).

Problem 76

Find exact sine, cosine, and tangent for 45 degrees using hypotenuse 2, each leg sqrt(2).

Open in simulator
Problem 77

Find exact sine, cosine, and tangent for 45 degrees using legs 2 and 2, hypotenuse 2sqrt(2).

Problem 78

Find exact sine, cosine, and tangent for 45 degrees using legs 3 and 3, hypotenuse 3sqrt(2).

Problem 79

Find exact sine, cosine, and tangent for 45 degrees using legs 1/2 and 1/2, hypotenuse sqrt(2)/2.

Problem 80

Find exact sine, cosine, and tangent for 45 degrees using hypotenuse 6, each leg 3sqrt(2).

Problem 81

Find exact sine, cosine, and tangent for 45 degrees using an isosceles right triangle with legs of length 7.

Problem 82

Find exact sine, cosine, and tangent for 45 degrees using a right triangle with angles 45, 45, and 90 degrees, and one leg of length 8.

Problem 83

Find exact sine, cosine, and tangent for 45 degrees using a 45-45-90 triangle with hypotenuse 10.

Problem 84

Find exact sine, cosine, and tangent for 45 degrees using a right triangle where the two acute angles are 45 degrees, and one leg is 'a'.

use special right triangle ratios.
12 problems Warmup Practice Mixed Review Assessment
Problem 85

Find exact sine, cosine, and tangent for 30 and 60 degrees using 30-60-90 with sides 1, sqrt(3), 2.

Problem 86

Find exact sine, cosine, and tangent for 30 and 60 degrees using short leg x, long leg xsqrt(3), hypotenuse 2x.

Problem 87

Find exact sine, cosine, and tangent for 30 and 60 degrees using hypotenuse 10, short leg 5, long leg 5sqrt(3).

Problem 88

Find exact sine, cosine, and tangent for 30 and 60 degrees using long leg sqrt(3), short leg 1.

Problem 89

Find exact sine, cosine, and tangent for 30 and 60 degrees using 30-60-90 triangle with short leg 2.

Problem 90

Find exact sine, cosine, and tangent for 30 and 60 degrees using A right triangle with angles 30 and 60 degrees and hypotenuse 6.

Problem 91

Find exact sine, cosine, and tangent for 30 and 60 degrees using A 30-60-90 triangle where the side opposite the 60-degree angle is 5sqrt(3).

Problem 92

Find exact sine, cosine, and tangent for 30 and 60 degrees using A 30-60-90 triangle with hypotenuse 1.

Open in simulator
Problem 93

Find exact sine, cosine, and tangent for 30 and 60 degrees using A right triangle with a 30-degree angle and the adjacent leg to 60-degrees is sqrt(3).

Problem 94

Find exact sine, cosine, and tangent for 30 and 60 degrees using A 30-60-90 triangle with the longer leg measuring 9 units.

Problem 95

Find exact sine, cosine, and tangent for 30 and 60 degrees using A 30-60-90 triangle where the side opposite the 30-degree angle is 1/2.

Problem 96

Find exact sine, cosine, and tangent for 30 and 60 degrees using A right triangle with acute angles 30 and 60 degrees, and hypotenuse 2sqrt(3).

model and solve exact side lengths.
12 problems Warmup Practice Mixed Review Assessment
Problem 97

Use special triangles in the applied context: A 30 degree ramp reaches a height of 6 ft.

Problem 98

Use special triangles in the applied context: An equilateral sign has side length 10 in; find its height.

Problem 99

Use special triangles in the applied context: A square has diagonal 8 cm; find its side length.

Problem 100

Use special triangles in the applied context: A 45 degree support brace reaches 7 ft up a wall.

Problem 101

Use special triangles in the applied context: A ladder leans against a wall, making a 60-degree angle with the ground. The ladder is 10 meters long.

Problem 102

Use special triangles in the applied context: A right isosceles triangle has one leg measuring 9 inches.

Problem 103

Use special triangles in the applied context: A kite string is 50 feet long and makes a 30-degree angle with the ground. Assume the string is straight.

Problem 104

Use special triangles in the applied context: A square garden has a diagonal path 12 feet long.

Problem 105

Use special triangles in the applied context: A pole casts a shadow 15 meters long when the angle of elevation to the sun is 60 degrees.

Problem 106

Use special triangles in the applied context: A path forms a right angle with a fence. A dog runs 8 yards along the path, then turns 45 degrees and runs to a tree. The path to the tree forms a 45-degree angle with the fence.

Open in simulator
Problem 107

Use special triangles in the applied context: An equilateral triangle has an altitude of 9 cm.

Problem 108

Use special triangles in the applied context: A square window pane has a side length of 11 inches.

identify angle pattern and apply matching ratio.
12 problems Warmup Practice Mixed Review Assessment
Problem 109

Choose between 30-60-90 and 45-45-90 structure for right triangle with one acute angle 45 degrees.

Problem 110

Choose between 30-60-90 and 45-45-90 structure for right triangle with one acute angle 30 degrees.

Problem 111

Choose between 30-60-90 and 45-45-90 structure for half of an equilateral triangle.

Problem 112

Choose between 30-60-90 and 45-45-90 structure for diagonal of a square creates a right triangle.

Open in simulator
Problem 113

Choose between 30-60-90 and 45-45-90 structure for an isosceles right triangle.

Problem 114

Choose between 30-60-90 and 45-45-90 structure for right triangle with one acute angle 60 degrees.

Problem 115

Choose between 30-60-90 and 45-45-90 structure for right triangle where one leg is half the hypotenuse.

Problem 116

Choose between 30-60-90 and 45-45-90 structure for right triangle with angles measuring 45, 45, and 90 degrees.

Problem 117

Choose between 30-60-90 and 45-45-90 structure for the triangle formed by drawing an altitude in an equilateral triangle.

Problem 118

Choose between 30-60-90 and 45-45-90 structure for right triangle with two equal acute angles.

Problem 119

Choose between 30-60-90 and 45-45-90 structure for right triangle with angles measuring 30, 60, and 90 degrees.

Problem 120

Choose between 30-60-90 and 45-45-90 structure for right triangle whose hypotenuse is sqrt(2) times the length of a leg.

reason about relative sine/cosine/tangent values.
12 problems Warmup Practice Mixed Review Assessment
Problem 121

Compare exact trig values from special triangles: sin30 and sin45.

Problem 122

Compare exact trig values from special triangles: cos30 and cos60.

Problem 123

Compare exact trig values from special triangles: tan45 and tan60.

Problem 124

Compare exact trig values from special triangles: sin30 and cos60.

Problem 125

Compare exact trig values from special triangles: sin60 and cos30.

Problem 126

Compare exact trig values from special triangles: tan30 and tan45.

Problem 127

Compare exact trig values from special triangles: cos45 and sin45.

Problem 128

Compare exact trig values from special triangles: sec60 and csc30.

Problem 129

Compare exact trig values from special triangles: cot30 and tan60.

Problem 130

Compare exact trig values from special triangles: cos60 and sin45.

Open in simulator
Problem 131

Compare exact trig values from special triangles: tan30 and cot60.

Problem 132

Compare exact trig values from special triangles: sec30 and sec45.

catch swapped sides, wrong radical, wrong angle, and rationalization mistakes.
12 problems Warmup Practice Mixed Review Assessment
Problem 133

Correct the special-right-triangle error: A student says a 45-45-90 hypotenuse is twice a leg.

Problem 134

Correct the special-right-triangle error: A student swaps the short and long legs in a 30-60-90 triangle.

Problem 135

Correct the special-right-triangle error: A student gives tan30 as sqrt(3).

Problem 136

Correct the special-right-triangle error: A student leaves a 45-degree sine value as 1/sqrt(2) when simplified exact form is requested.

Problem 137

Correct the special-right-triangle error: A student states that in a 30-60-90 triangle, the hypotenuse is sqrt(3) times the short leg.

Problem 138

Correct the special-right-triangle error: A student says the long leg in a 30-60-90 triangle is twice the short leg.

Open in simulator
Problem 139

Correct the special-right-triangle error: A student calculates sin60 degrees as 1/2.

Problem 140

Correct the special-right-triangle error: A student gives cos30 degrees as 1/2.

Problem 141

Correct the special-right-triangle error: A student states tan45 degrees is sqrt(2).

Problem 142

Correct the special-right-triangle error: A student incorrectly states that in a 45-45-90 triangle, the legs are in a 1:sqrt(2) ratio.

Problem 143

Correct the special-right-triangle error: A student finds the short leg of a 30-60-90 triangle by dividing the hypotenuse by sqrt(3).

Problem 144

Correct the special-right-triangle error: A student calculates a side length as 6/sqrt(3) and leaves it unsimplified.