True or False? The graph of sin x passes through the origin .

True. The graph of sin x passes through the origin .

Define the range of sin x and cos x .

The range of both sin x and cos x is -1 ≤ y ≤ 1.

What is the relationship between sin(- x ) and sin( x ) ?

The relationship between sin(- x ) and sin( x ) is sin(- x ) = -sin( x ) .

What is the relationship between cos(- x ) and cos( x ) ?

The relationship between cos(- x ) and cos( x ) is cos(- x ) = cos( x ) .

True or False? The order of applying transformations doesn't matter .

False. The order in which transformations are applied is important .

In what order should transformations be applied ?

Any stretches should be applied first, followed by any translations . Reflections should be applied last.

What type of phenomena can be modelled using trigonometric functions ?

Any phenomena that fluctuates periodically can be modelled using a trigonometric function. E.g. the vertical height above the ground of a person on a Ferris wheel over time, or the water depth of a tidal river over time, could be modelled using a trig function.

True or False? In real-life scenarios modelled by trigonometric functions, the time to complete a cycle always remains constant .

False. In real-life scenarios modelled by trigonometric functions, the time to complete a cycle does not always remain constant . The time taken to complete a cycle may change over time. This is one of the limitations of a trigonometric model.

True or False? One limitation of a trigonometric model is that the the amplitude is constant .

True. One limitation of a trigonometric model is that the the amplitude is constant . In real life, the amplitude is not always constant. For example, the function may get closer to the principal axis over time.

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Trigonometric Functions & Graphs (DP IB Analysis & Approaches (AA))

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FrontGraphs of Trigonometric Functions
Which function is this the graph of?
BackGraphs of Trigonometric Functions
The graph is of the tangent function.
It is a periodic graph with the following features:
x-intercepts at -180º, 0º, 180º, 360º (and every 180º in either direction),
a y-intercept at (0, 0),
asymptotes at -90º, 90º, 270º (and every 180º in either direction),
and a range of $y \in ℝ$ .
FrontGraphs of Trigonometric Functions
Which function is this the graph of?
BackGraphs of Trigonometric Functions
The graph is of the cosine function.
It is a periodic graph with the following features:
x-intercepts at -90º, 90º, 270º (and every 180º in either direction),
a y-intercept at (0, 1),
and a range of -1 ≤ y ≤ 1.
FrontGraphs of Trigonometric Functions
Which function is this the graph of?
BackGraphs of Trigonometric Functions
The graph is of the sine function.
It is a periodic graph with the following features:
x-intercepts at -180º, 0º, 180º, 360º (and every 180º in either direction),
a y-intercept at (0, 0),
and a range of -1 ≤ y ≤ 1.

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Which function is this the graph of?
The graph is of the tangent function.
It is a periodic graph with the following features:
- x-intercepts at -180º, 0º, 180º, 360º (and every 180º in either direction),
- a y-intercept at (0, 0),
- asymptotes at -90º, 90º, 270º (and every 180º in either direction),
- and a range of $y \in ℝ$ .
Which function is this the graph of?
The graph is of the cosine function.
It is a periodic graph with the following features:
- x-intercepts at -90º, 90º, 270º (and every 180º in either direction),
- a y-intercept at (0, 1),
- and a range of -1 ≤ y ≤ 1.
Which function is this the graph of?
The graph is of the sine function.
It is a periodic graph with the following features:
- x-intercepts at -180º, 0º, 180º, 360º (and every 180º in either direction),
- a y-intercept at (0, 0),
- and a range of -1 ≤ y ≤ 1.
True or False?
The graph of sin x passes through the origin.
True.
The graph of sin x passes through the origin.
What is the period of tan x?
The period of tan x is 180º (or $π$ radians).
What is the period of sin x and cos x?
The period of both sin x and cos x is 360º (or 2 $π$ radians).
Define the range of sin x and cos x.
The range of both sin x and cos x is -1 ≤ y ≤ 1.
Define the range of tan x.
The range of tan x is $y \in ℝ$ . I.e., tan x can take any real number value (positive, negative or zero).
What are the equations of the asymptotes in the graph of tan x, $- π \leq x \leq 2 π$ .
The equations of the asymptotes in the graph of tan x, $- π \leq x \leq 2 π$ are
$x = - \frac{π}{2}$ , $x = \frac{π}{2}$ and $x = \frac{3 π}{2}$ .
What is the relationship between sin(-x) and sin(x)?
The relationship between sin(-x) and sin(x) is sin(-x) = -sin(x).
What is the relationship between cos(-x) and cos(x)?
The relationship between cos(-x) and cos(x) is cos(-x) = cos(x).
True or False?
tan(x) = tan(x ± 180°)
True.
tan(x) = tan(x ± 180°) or tan(x ± $π$ )
What is the relationship between sin(x) and sin( $π$ - x)?
The relationship between sin(x) and sin( $π$ - x), is sin(x) = sin( $π$ - x), (or sin(x) = sin(180° - x) ).
What does $y = \sin (x) + a$ represent?
$y = \sin (x) + a$ represents a vertical translation of $y = \sin (x)$ by $a$ units in the positive y-direction.
How is a horizontal translation to the left by $a$ units represented for $\sin (x)$ ?
For $\sin (x)$ , a horizontal translation to the left by $a$ units is represented by $\sin (x + a)$ .
What transformation does $y = a \sin (x)$ represent?
$y = a \sin (x)$ represents a transformation of $y = \sin (x)$ by a vertical stretch with scale factor $a$ .
How is a horizontal stretch with scale factor $a$ represented for $\sin (x)$ ?
For $\sin (x)$ , a horizontal stretch with scale factor $a$ is represented by $y = \sin (\frac{x}{a})$ .
What transformation does $y = - \sin (x)$ represent?
$y = - \sin (x)$ represents a reflection of $\sin (x)$ about the x-axis.
In the function $a \sin (b x)$ , what does the absolute value of $a$ represent?
In the function $a \sin (b x)$ , the absolute value of $a$ represents the amplitude of the graph.
What is the period of $a \sin (b x)$ in degrees?
The period of $a \sin (b x)$ in degrees is $\frac{360 º}{b}$ .
In $a \sin (b (x - c)) + d$ , what does $c$ represent?
In $a \sin (b (x - c)) + d$ , the $c$ represents a horizontal translation.
What is the equation of the principal axis for the function $a \sin (b (x - c)) + d$ ?
The equation of the principal axis for the function $a \sin (b (x - c)) + d$ is $y = d$ .
What is the period of $a \tan (b (x - c)) + d$ in radians?
The period of $a \tan (b (x - c)) + d$ is $\frac{π}{b}$ radians.
True or False?
The order of applying transformations doesn't matter.
False.
The order in which transformations are applied is important.
In what order should transformations be applied?
Any stretches should be applied first, followed by any translations. Reflections should be applied last.
What line will the maximum points for $a \sin (b (x - c)) + d$ lie on?
The maximum points for $a \sin (b (x - c)) + d$ will lie on the line $y = a + d$ .
What type of phenomena can be modelled using trigonometric functions?
Any phenomena that fluctuates periodically can be modelled using a trigonometric function.
E.g. the vertical height above the ground of a person on a Ferris wheel over time, or the water depth of a tidal river over time, could be modelled using a trig function.
If a trigonometric function $y = a \sin (b (x - c)) + d$ is used to model the water depth of a tidal river, what would $a$ represent?
In a trigonometric model $y = a \sin (b (x - c)) + d$ , the absolute value of $a$ represents the amplitude of the function.
If the function is used to model the water depth of a tidal river, $a$ would represent the maximum distance that the water depth would be above and below the principal axis.
In the trigonometric function $y = a \sin (b (x - c)) + d$ , how does the value of $b$ affect the model?
In the trigonometric function $y = a \sin (b (x - c)) + d$ , changing the value of $b$ changes the period of the function. The bigger the value of $b$ , the quicker the function repeats a cycle.
In the trigonometric function $y = a \sin (b (x - c)) + d$ , how does the value of $d$ affect the model?
In the trigonometric function $y = a \sin (b (x - c)) + d$ , $y = d$ is the principal axis.
As the value of $d$ increases, the graph is shifted up, as the value of $d$ decreases, the graph is shifted down.
In the trigonometric function $y = a \sin (b (x - c)) + d$ , what does $c$ represent?
In the trigonometric function $y = a \sin (b (x - c)) + d$ , the horizontal shift of the graph is represented by $c$ .
True or False?
In real-life scenarios modelled by trigonometric functions, the time to complete a cycle always remains constant.
False.
In real-life scenarios modelled by trigonometric functions, the time to complete a cycle does not always remain constant.
The time taken to complete a cycle may change over time.
This is one of the limitations of a trigonometric model.
True or False?
One limitation of a trigonometric model is that the the amplitude is constant.
True.
One limitation of a trigonometric model is that the the amplitude is constant.
In real life, the amplitude is not always constant.
For example, the function may get closer to the principal axis over time.

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Trigonometric Functions & Graphs (DP IB Analysis & Approaches (AA))

Flashcards

Cards in this collection (33)

1. Number & Algebra

Number & Algebra Toolkit

Exponentials & Logs

Sequences & Series

Simple Proof & Reasoning

Proof by Induction & Contradiction

Binomial Theorem

Permutations & Combinations

Complex Numbers

Further Complex Numbers

Systems of Linear Equations

2. Functions

Linear Functions & Graphs

Quadratic Functions & Graphs

Functions Toolkit

Other Functions & Graphs

Reciprocal & Rational Functions

Transformations of Graphs

Polynomial Functions

Inequalities

Modulus Functions & Further Transformations

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

The Unit Circle & Exact Values

Trigonometric Functions & Graphs

Trigonometric Equations & Identities

Inverse & Reciprocal Trigonometric Functions

Trigonometric Proof & Equation Strategies

Vector Properties

Vector Equations of Lines

Vector Planes

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Probability

Discrete Random Variables

Binomial Distribution

Normal Distribution

Continuous Random Variables

5. Calculus

Differentiation

Techniques & Applications of Differentiation

Integration

Techniques & Applications of Integration

Optimisation

Kinematics

Basic Limits & Continuity

Further Differentiation

Further Integration

Differential Equations

Maclaurin Series

Limits using l'Hôpital's Rule & Maclaurin Series