RATIOS OF COMPLEMENTARY ANGLES AND TRIGONOMETRIC CO-FUNCTIONS

Lesson 2 (Seah Shao Xuan, Tan Yong Yi, Vincent Chung, Quek Yong Jie) 3rd May 2011

Part 1

RECAP

HYPOTENUSE, ADJACENT AND OPPOSITE The adjacent, opposite and hypotenuse are the 3

different sides we can find on a right-angled triangle.

The hypotenuse is the longest side of the triangle. The angle measured determines the position of

the adjacent and opposite sides. The adjacent side is the line touching the angle

other than the hypotenuse. The opposite side is the line which the angle does

not use.hypotenus

eadjacent

opposite

SINE, COSINE AND TANGENT Sine, cosine and tangent are the 3 basic

functions of calculating the ratio of two different sides of a right-angled triangle by inputting its angle.

Sine is defined by the opposite over the hypotenuse, cosine is defined by the adjacent over the hypotenuse and the tangent is defined by the opposite over the adjacent.

Sine, cosine and tangent is usually spelt as sin, cos and tan respectively when in math sums.

Part 2

TRIGONOMETRIC CO-FUNCTIONS

COSECANT The cosecant (or csc) is the reciprocal of the

sine function. = Thus, as the sine is , the cosecant is . When we use cosecant for the angle of a right-

angled triangle, the ratio we get is the hypotenuse over the opposite.

That means that the value we get is the unit length of the hypotenuse for 1 unit length on the opposite.

SECANT The secant (or sec) is the reciprocal of the

cosine function. = Thus, as the sine is , the cosecant is . When we use secant for the angle of a right-

angled triangle, the ratio we get is the hypotenuse over the adjacent.

That means that the value we get is the unit length of the hypotenuse for 1 unit length on the adjacent.

COTANGENT The cotangent (or cot) is the reciprocal of the

tangent function. = Thus, as the tangent is , the cosecant is . When we use cotangent for the angle of a

right-angled triangle, the ratio we get is the adjacent over the opposite.

That means that the value we get is the unit length of the adjacent for 1 unit length on the opposite. This is also known as run over rise.

SECANT, COSECANT AND COTANGENT

sec =

csc =

cot =

Part 2

COMPLEMENTARY ANGLES

COMPLEMENTARY ANGLES Complementary angles are two angles

that make up a total of 90°. The complementary angle of a right-

angled triangle is the last angle, other than the right-angle and the designated angle.

hypo

tenus

e

adjacent

opposite

Part 3a

SINE AND COSINE: RELATIONSHIPS

FUNCTIONS OF COMPLEMENTARY ANGLES (SINE AND COSINE)

hypotenuse (x)

aadjacent (z)

opposite (y)1

sin(a) = =

FUNCTIONS OF COMPLEMENTARY ANGLES (SINE AND COSINE)

hypotenuse (x)

b

opposite (z)

adjacent (y)2

cos(b) = =

FUNCTIONS OF COMPLEMENTARY ANGLES (SINE AND COSINE)

3

cos(b) = = sin(a) = =

x

az

yb

sin(a) = cos(b)

Part 3b

SECANT AND COSECANT: RELATIONSHIPS

FUNCTIONS OF COMPLEMENTARY ANGLES (SECANT AND COSECANT)

hypotenuse (x)

aadjacent (z)

opposite (y)1

sec(a) = =

FUNCTIONS OF COMPLEMENTARY ANGLES (SECANT AND COSECANT)

hypotenuse (x)

b

opposite (z)

adjacent (y)2

csc(b) = =

csc(b) = = sec(a) = =

FUNCTIONS OF COMPLEMENTARY ANGLES (SECANT AND COSECANT)

3 x

az

yb

sec(a) = csc(b)

Part 3c

TANGENT AND COTANGENT: RELATIONSHIPS

FUNCTIONS OF COMPLEMENTARY ANGLES (TANGENT AND COTANGENT)

hypotenuse (x)

aadjacent (z)

opposite (y)1

tan(a) = =

FUNCTIONS OF COMPLEMENTARY ANGLES (TANGENT AND COTANGENT)

hypotenuse (x)

b

opposite (z)

adjacent (y)2

cot(b) = =

csc(b) = = tan(a) = =

FUNCTIONS OF COMPLEMENTARY ANGLES (TANGENT AND COTANGENT)

3 x

az

yb

tan(a) = cot(b)

RELATIONSHIPS (ALL SUMMED UP IN A TABLE)

Ratio of Angle (Θ)

Ratio of complementary angle

(δ)Relationshi

p 1 Relationship 2

sinΘ = cosδ = ( if were non-com.) sinΘ = cosδ sinΘ = cos(90 -

Θ)

cosΘ = sinδ = ( if were non-com.) cosΘ = sinδ cosΘ = sin(90 -

Θ)

tanΘ = cotδ = ( if were non-com.) tanΘ = cotδ tanΘ = cot(90 -

Θ)

cotΘ = tanδ = ( if were non-com.) cotΘ = tanδ cotΘ = tan(90 -

Θ)

secΘ = cscδ = ( if were non-com.) secΘ = cscδ secΘ = csc(90 -

Θ)

cscΘ = secδ = ( if were non-com.) cscΘ = secδ cscΘ = sec(90 -

Θ)

Part 4

TAKE NOTE

THINGS TO TAKE NOTE When we measure the angle, if it is

measured anticlockwise from the right, it is positive.

If the angle is measured clockwise from the right, it is negative.

This angle can affect your results!

The End