Mathspresso
Math II · G-SRT.7

Explaining and Using the Complementary-Angle Relationship Between Sine and Cosine

Sine and cosine are not random buttons on a calculator; they are paired ways of measuring the same right-triangle shape from opposite acute angles.

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 acute angles sum to 90 degrees.
12 problems Warmup Practice Mixed Review Assessment
Problem 1

Identify complementary acute angles in a right triangle from one acute angle is 35 degrees.

Problem 2

Identify complementary acute angles in a right triangle from angles are x and 2x in a right triangle.

Problem 3

Identify complementary acute angles in a right triangle from one acute angle is a degrees.

Problem 4

Identify complementary acute angles in a right triangle from acute angles are 4y and 5y.

Problem 5

Identify complementary acute angles in a right triangle from one acute angle is 20 degrees.

Problem 6

Identify complementary acute angles in a right triangle from one acute angle is 45 degrees.

Problem 7

Identify complementary acute angles in a right triangle from one acute angle is (x + 15) degrees.

Problem 8

Identify complementary acute angles in a right triangle from acute angles are (2x - 5) and (3x + 10) degrees.

Problem 9

Identify complementary acute angles in a right triangle from the ratio of the two acute angles is 1:2.

Problem 10

Identify complementary acute angles in a right triangle from one acute angle is (2y - 10) degrees.

Problem 11

Identify complementary acute angles in a right triangle from one acute angle is 10 degrees greater than the other.

Problem 12

Identify complementary acute angles in a right triangle from acute angles are (x/2 + 5) and (x + 10) degrees.

Open in simulator
apply cofunction relationship.
12 problems Warmup Practice Mixed Review Assessment
Problem 13

Use sin(theta)=cos(90-theta) for 30 degrees.

Problem 14

Use sin(theta)=cos(90-theta) for 45 degrees.

Problem 15

Use sin(theta)=cos(90-theta) for 12 degrees.

Problem 16

Use sin(theta)=cos(90-theta) for x degrees.

Problem 17

Use sin(theta)=cos(90-theta) for 60 degrees.

Problem 18

Use sin(theta)=cos(90-theta) for 75 degrees.

Problem 19

Use sin(theta)=cos(90-theta) for 1 degree.

Problem 20

Use sin(theta)=cos(90-theta) for 89 degrees.

Problem 21

Use sin(theta)=cos(90-theta) for 25 degrees.

Problem 22

Use sin(theta)=cos(90-theta) for 50 degrees.

Open in simulator
Problem 23

Use sin(theta)=cos(90-theta) for y degrees.

Problem 24

Use sin(theta)=cos(90-theta) for (2x) degrees.

apply cofunction relationship.
12 problems Warmup Practice Mixed Review Assessment
Problem 25

Use cos(theta)=sin(90-theta) for 60 degrees.

Problem 26

Use cos(theta)=sin(90-theta) for 45 degrees.

Problem 27

Use cos(theta)=sin(90-theta) for 18 degrees.

Problem 28

Use cos(theta)=sin(90-theta) for x degrees.

Problem 29

Use cos(theta)=sin(90-theta) for 30 degrees.

Problem 30

Use cos(theta)=sin(90-theta) for 75 degrees.

Problem 31

Use cos(theta)=sin(90-theta) for 1 degree.

Problem 32

Use cos(theta)=sin(90-theta) for 89 degrees.

Problem 33

Use cos(theta)=sin(90-theta) for 22.5 degrees.

Problem 34

Use cos(theta)=sin(90-theta) for y degrees.

Open in simulator
Problem 35

Use cos(theta)=sin(90-theta) for (2x) degrees.

Problem 36

Use cos(theta)=sin(90-theta) for (a+b) degrees.

show opposite/adjacent swap between complementary angles.
12 problems Warmup Practice Mixed Review Assessment
Problem 37

Explain sine-cosine complementarity from side labels for right triangle with acute angles A and B.

Problem 38

Explain sine-cosine complementarity from side labels for 30-60-90 triangle.

Problem 39

Explain sine-cosine complementarity from side labels for right triangle with complementary angles theta and phi.

Problem 40

Explain sine-cosine complementarity from side labels for any right triangle.

Problem 41

Explain sine-cosine complementarity from side labels for right triangle PQR with right angle Q.

Problem 42

Explain sine-cosine complementarity from side labels for a right triangle with acute angles alpha and beta.

Problem 43

Explain sine-cosine complementarity from side labels for a right triangle with legs 'a' and 'b', and hypotenuse 'c'.

Open in simulator
Problem 44

Explain sine-cosine complementarity from side labels for a right triangle with complementary acute angles X and Y.

Problem 45

Explain sine-cosine complementarity from side labels for right triangle DEF with right angle E.

Problem 46

Explain sine-cosine complementarity from side labels for any right triangle with acute angles M and N.

Problem 47

Explain sine-cosine complementarity from side labels for a right triangle where one acute angle is 40 degrees.

Problem 48

Explain sine-cosine complementarity from side labels for a right triangle with acute angles whose sum is 90 degrees.

set angles complementary.
12 problems Warmup Practice Mixed Review Assessment
Problem 49

Find the missing angle from equal sine/cosine expressions sin(3x°)=cos(60°).

Problem 50

Find the missing angle from equal sine/cosine expressions sin((x+20)°)=cos(40°).

Problem 51

Find the missing angle from equal sine/cosine expressions cos(2y°)=sin(30°).

Problem 52

Find the missing angle from equal sine/cosine expressions sin(5a°)=cos((2a+20)°).

Problem 53

Find the missing angle from equal sine/cosine expressions sin(x°)=cos(70°).

Problem 54

Find the missing angle from equal sine/cosine expressions cos(y°)=sin(55°).

Open in simulator
Problem 55

Find the missing angle from equal sine/cosine expressions sin((2x+10)°)=cos(30°).

Problem 56

Find the missing angle from equal sine/cosine expressions cos((3y-5)°)=sin(35°).

Problem 57

Find the missing angle from equal sine/cosine expressions sin((4m)°)=cos((m+10)°).

Problem 58

Find the missing angle from equal sine/cosine expressions cos((2p+5)°)=sin((3p-15)°).

Problem 59

Find the missing angle from equal sine/cosine expressions sin((x/2+10)°)=cos(50°).

Problem 60

Find the missing angle from equal sine/cosine expressions cos((4k-20)°)=sin((k+5)°).

transform trig notation.
12 problems Warmup Practice Mixed Review Assessment
Problem 61

Rewrite sine expression sin(25 degrees) as cosine of its complement.

Problem 62

Rewrite sine expression sin(70 degrees) as cosine of its complement.

Problem 63

Rewrite sine expression sin(x degrees) as cosine of its complement.

Problem 64

Rewrite sine expression sin(2a+10 degrees) as cosine of its complement.

Problem 65

Rewrite sine expression sin(45 degrees) as cosine of its complement.

Problem 66

Rewrite sine expression sin(10 degrees) as cosine of its complement.

Problem 67

Rewrite sine expression sin(30.5 degrees) as cosine of its complement.

Problem 68

Rewrite sine expression sin(15.25 degrees) as cosine of its complement.

Problem 69

Rewrite sine expression sin(3y degrees) as cosine of its complement.

Open in simulator
Problem 70

Rewrite sine expression sin(b-5 degrees) as cosine of its complement.

Problem 71

Rewrite sine expression sin(3m+20 degrees) as cosine of its complement.

Problem 72

Rewrite sine expression sin(45-z degrees) as cosine of its complement.

transform trig notation.
12 problems Warmup Practice Mixed Review Assessment
Problem 73

Rewrite cosine expression cos(25 degrees) as sine of its complement.

Problem 74

Rewrite cosine expression cos(70 degrees) as sine of its complement.

Problem 75

Rewrite cosine expression cos(x degrees) as sine of its complement.

Problem 76

Rewrite cosine expression cos(3a degrees) as sine of its complement.

Problem 77

Rewrite cosine expression cos(10 degrees) as sine of its complement.

Problem 78

Rewrite cosine expression cos(45 degrees) as sine of its complement.

Open in simulator
Problem 79

Rewrite cosine expression cos(85 degrees) as sine of its complement.

Problem 80

Rewrite cosine expression cos(2y degrees) as sine of its complement.

Problem 81

Rewrite cosine expression cos((x+10) degrees) as sine of its complement.

Problem 82

Rewrite cosine expression cos((50-b) degrees) as sine of its complement.

Problem 83

Rewrite cosine expression cos(theta degrees) as sine of its complement.

Problem 84

Rewrite cosine expression cos((2x+5) degrees) as sine of its complement.

reason about equivalent ratios.
12 problems Warmup Practice Mixed Review Assessment
Problem 85

Use complementary relationship to compare trig values sin(35) and cos(55).

Problem 86

Use complementary relationship to compare trig values cos(20) and sin(70).

Problem 87

Use complementary relationship to compare trig values sin(40) and cos(40).

Problem 88

Use complementary relationship to compare trig values sin(12) and cos(78).

Problem 89

Use complementary relationship to compare trig values sin(10) and cos(80).

Problem 90

Use complementary relationship to compare trig values sin(45) and cos(45).

Problem 91

Use complementary relationship to compare trig values sin(30) and cos(30).

Problem 92

Use complementary relationship to compare trig values sin(1) and cos(89).

Problem 93

Use complementary relationship to compare trig values sin(60) and cos(30).

Problem 94

Use complementary relationship to compare trig values sin(50) and cos(20).

Open in simulator
Problem 95

Use complementary relationship to compare trig values cos(25) and sin(65).

Problem 96

Use complementary relationship to compare trig values cos(70) and sin(10).

choose alternate angle to simplify side relationship.
12 problems Warmup Practice Mixed Review Assessment
Problem 97

Use cofunction relationship in right-triangle context angle of elevation is 35 degrees and the other acute angle is needed.

Problem 98

Use cofunction relationship in right-triangle context angle of depression forms complement with 62 degrees.

Problem 99

Use cofunction relationship in right-triangle context right triangle has acute angles theta and phi.

Problem 100

Use cofunction relationship in right-triangle context sightline triangle uses angle 20 degrees at the ground.

Problem 101

Use cofunction relationship in right-triangle context A right triangle has one acute angle measuring 40 degrees.

Problem 102

Use cofunction relationship in right-triangle context The angle formed by a ramp with the horizontal is 25 degrees.

Problem 103

Use cofunction relationship in right-triangle context In a right triangle, the two acute angles are labeled 'alpha' and 'beta'.

Problem 104

Use cofunction relationship in right-triangle context A roof pitch creates an angle of 30 degrees with the horizontal in a right-angled cross-section.

Problem 105

Use cofunction relationship in right-triangle context One acute angle in a right triangle is 72 degrees.

Open in simulator
Problem 106

Use cofunction relationship in right-triangle context The angle of a shadow cast by a pole is 50 degrees from the ground.

Problem 107

Use cofunction relationship in right-triangle context A right triangle has acute angles 'x' and 'y'.

Problem 108

Use cofunction relationship in right-triangle context A right triangle has an acute angle of 45 degrees.

catch non-complementary angles and tangent confusion.
12 problems Warmup Practice Mixed Review Assessment
Problem 109

Identify invalid cofunction statement sin(30)=cos(60).

Problem 110

Identify invalid cofunction statement sin(30)=cos(30).

Problem 111

Identify invalid cofunction statement cos(75)=sin(15).

Problem 112

Identify invalid cofunction statement tan(20)=cos(70).

Problem 113

Identify invalid cofunction statement sin(45)=cos(50).

Problem 114

Identify invalid cofunction statement tan(10)=cot(80).

Problem 115

Identify invalid cofunction statement cos(10)=sin(90).

Open in simulator
Problem 116

Identify invalid cofunction statement sec(15)=csc(75).

Problem 117

Identify invalid cofunction statement sin(10)=tan(80).

Problem 118

Identify invalid cofunction statement tan(60)=cot(20).

Problem 119

Identify invalid cofunction statement sec(40)=sin(50).

Problem 120

Identify invalid cofunction statement sin(89)=cos(1).

compare ratio structures.
12 problems Warmup Practice Mixed Review Assessment
Problem 121

Explain why tangent does not have the same sine-cosine complement relationship for theta and 90-theta.

Problem 122

Explain why tangent does not have the same sine-cosine complement relationship for 30 and 60 degrees.

Problem 123

Explain why tangent does not have the same sine-cosine complement relationship for two acute complementary angles.

Problem 124

Explain why tangent does not have the same sine-cosine complement relationship for angle A and complement B.

Problem 125

Explain why tangent does not have the same sine-cosine complement relationship for alpha and 90-alpha.

Problem 126

Explain why tangent does not have the same sine-cosine complement relationship for 15 and 75 degrees.

Problem 127

Explain why tangent does not have the same sine-cosine complement relationship for the acute angles of a right triangle.

Problem 128

Explain why tangent does not have the same sine-cosine complement relationship for x and 90-x.

Open in simulator
Problem 129

Explain why tangent does not have the same sine-cosine complement relationship for 25 and 65 degrees.

Problem 130

Explain why tangent does not have the same sine-cosine complement relationship for an angle and its complement.

Problem 131

Explain why tangent does not have the same sine-cosine complement relationship for angle A and (90-A).

Problem 132

Explain why tangent does not have the same sine-cosine complement relationship for two angles that sum to 90 degrees.

catch wrong complement, swapped function, and degree arithmetic mistakes.
12 problems Warmup Practice Mixed Review Assessment
Problem 133

Correct the complementary-angle trig error: A student says sin(30)=cos(30).

Problem 134

Correct the complementary-angle trig error: A student rewrites cos(20) as sin(80).

Problem 135

Correct the complementary-angle trig error: A student sets sin(x)=cos(2x) by x=2x.

Problem 136

Correct the complementary-angle trig error: A student uses 180-theta as the complement.

Problem 137

Correct the complementary-angle trig error: A student claims tan(25) is the same as cot(25).

Problem 138

Correct the complementary-angle trig error: A student writes sec(40) = csc(40).

Problem 139

Correct the complementary-angle trig error: When solving sin(4x) = cos(2x), a student sets 4x = 2x.

Problem 140

Correct the complementary-angle trig error: A student thinks the cofunction of sin(theta) is tan(90-theta).

Open in simulator
Problem 141

Correct the complementary-angle trig error: A student uses 180-70 to find the cofunction angle for cos(70).

Problem 142

Correct the complementary-angle trig error: A student rewrites csc(80) as sec(80).

Problem 143

Correct the complementary-angle trig error: A student states that if sin(A) = cos(B), then A must equal B.

Problem 144

Correct the complementary-angle trig error: A student says the complement of 15 degrees is 70 degrees.