Every equation has a specific graph. For example, y = x + 2 is an equation that has a specific graph and if we change any part of the equation it will result in a new graph and trigonometric functions are no exception. These functions also have a specific graph and like others, if you change their parameter, it will result in a different graph. That graph can be similar to the original graph and it cannot be. For example, we have two equations y = \cos { x }, and y = \cos { 2x }, both equations have the same trigonometric functions but the only difference is that they both have different angles. The graph of both equations will be different but they will be similar to each other.

Before we start about different trigonometric functions, you should know all the elements of the trigonometric function. For example, you have y = \sin { 3x }, the

Sine Function

y = sin x

Domain: R

Range: [-1, 1]

Period: 2 \pi \quad rad

Continuity: continuous at \forall \times \in R

Increasing: ... U (-\frac { \pi }{ 2 }, \frac { \pi }{ 2 }) U ( \frac { 3 \pi }{ 2 }, \frac { 5 \pi }{ 2 }) U ...

Decreasing: ... U (\frac { \pi }{ 2 }, \frac { 3 \pi }{ 2 }) U ( \frac { 5 \pi }{ 2 }, \frac { 7 \pi }{ 2 }) U ...

Maximum: (\frac { \pi }{ 2 } + 2 \pi . k, 1) \qquad k \in Z

Minimum: (\frac { 3 \pi }{ 2 } + 2 \pi . k, -1) \qquad k \in Z

Odd function: \sin { (-x) } = - \sin { x }

x-intercepts: x = {0 + \pi . k }

 

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Cosine Function

y = cos x

Domain: R

Range: [-1, 1]

Period: 2 \pi \quad rad

Continuity: continuous at \forall \times \in R

Increasing: ... U (- \pi, 0) U ( \pi, 2 \pi) U ...

Decreasing: ... U (0, \pi) U ( 2 \pi, 3 \pi) U ...

Maximum: (2 \pi + k, 1) \qquad k \in Z

Minimum: ( \pi . (2k + 1), -1) \qquad k \in Z

Even function: \cos { (-x) } = \cos { x }

x-intercepts:   x = { \frac { \pi }{ 2 } + k }

 

Tangent Function

y = tan x

Domain: R - {(2k + 1) . \frac { \pi }{ 2 }, k \in Z} = R - {..., - \frac { \pi }{ 2 }, \frac { \pi }{ 2 }, \frac { 3 \pi }{ 2 }, ...}

Range: R

Continuity: continuous at \forall \times \in R - {( \frac { \pi }{ 2 } + \pi . k)}

Period: \pi rad

Increasing: R

Maximum: No.

Minimum: No.

Odd function: \tan { -x } = - \tan { x }

x-intercepts:  x = { 0 + \pi . k }

 

Cotangent Function

y = cot x

Domain: R - {k . \pi, k \in Z} = R - {..., - \pi, 0, \pi, ...}

Range: R

Continuity: continuous at \times \in R - {( \pi . k , k \in Z )}

Period: \pi rad

Decreasing: R

Maximum: No.

Minimum: No.

Odd function: \cot { -x } = - \cot { x }

x-intercepts:   x = { \frac { \pi }{ 2 } + k }

 

Secant Function

y = sec x

Domain: R - { (2k + 1) . \frac { \pi }{ 2 }, k \in Z } = R - {..., - \frac { \pi }{ 2 }, \frac { \pi }{ 2 }, \frac { 3 \pi }{ 2 }, ...}

Range: ( -\infty , -1] U [ 1, \infty)

Period: 2 \pi rad

Continuity: continuous at \forall \times \in R - {( \frac { \pi }{ 2 } + \pi . k)}

Increasing: ... U(0, \frac { \pi }{ 2 }) U ( \frac { \pi }{ 2 }, \pi) U ...

Decreasing: ... U (\pi, \frac { 3 \pi }{ 2 }) U ( \frac { 3 \pi }{ 2 }, 2 \pi) U ...

Maximum: (2 \pi . k, -1) \qquad k \in Z

Minimum: (\pi . (2k + 1), -1) \qquad k \in Z

Even function: \sec { -x } = \sec { x }

x-intercepts:    No

 

Cosecant Function

y = csc x

Domain: R - {k . \pi, k \in Z} = R - {..., - \pi, 0, \pi, ...}

Range: ( - \infty, -1] U [1, \infty )

Period: 2 \pi rad

Continuity: continuous at \times \in R - { \pi . k, k \in Z }

Increasing: ... U (\frac { \pi }{ 2 }, \pi) U (\pi, \frac { 3 \pi }{ 2 }) U ...

Decreasing: ... U (0, \frac { \pi }{ 2 }) U (\frac { 3 \pi }{ 2 }, 2 \pi) U...

Maximum: (\frac { 3 \pi }{ 2 } + 2 \pi . k, -1) \qquad k \in Z

Minimum: (\frac { \pi }{ 2 } + 2 \pi . k, -1) \qquad k \in Z

Odd function: \cosec { -x } = -\cosec { x }

x-intercepts:    No

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.