1.1 Solving and Simplifying Inverse Trigonometric Functions using Right Triangle Method

Foundation of Applied Mathematics
(MAT438)

MOOC MAT438/ UiTM

Foundation of Applied Mathematics
(MAT438)

Chapter 1 Inverse Trigonometric Functions
Chapter 2 Hyperbolic and Inverse Hyperbolic functions
Chapter 3 Integration Techniques
Chapter 4 Ordinary Differential Equation (ODE)

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Chapter 1 : Inverse Trigonometric Functions

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Inverse Trigonometric Functions

Inverse Trigonometric Functions

① Introduction (notation, properties and their graphs)
② Evaluating Inverse Trigonometric Functions using Triangle Method
③ Derivatives of Inverse Trigonometric Functions
④ Integration of Inverse Trigonometric Functions

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Introduction

Each of the trigonometric functions sine, cosine, tangent, secant,
cosecant and cotangent has an inverse (with a restricted domain)
The inverse is used to obtain the measure of an angle using the
ratios from basic right triangle trigonometry.
For example, the inverse of sine is denoted as  sin-1 or Arcsine or
asin.

Note: this does NOT mean sine to the power of negative one.

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Inverse Trigonometric Functions Other Names

Notation Sometimes sin-1 is called asin or arcsin
Likewise cos-1 is called acos or arccos
MOOC MAT438/ UiTM And tan-1 is called atan or arctan

Inverse Trigonometric Functions

Properties of Inverse Trigonometric Functions

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Inverse Trigonometric Functions

Graph of Inverse Trigonometric Functions

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Inverse Trigonometric Functions

Coming up next…

Procedure & Useful Formula to Evaluate
Inverse Trigonometric Functions
using Triangle Method

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MOOC MAT438/ UiTM

Inverse Trigonometric Functions

 Introduction (notation, Properties and their graphs)
 Evaluating Inverse Trigonometric Functions using Triangle Method
 Derivatives of Inverse Trigonometric Functions
 Integration of Inverse Trigonometric Functions

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Procedure to Evaluate Inverse Trigonometric Function Using Triangle Method
Step 1 Let A  inverse trigonometric function

(or A  first inverse trigonometric function, and
B  second inverse trigonometric function

Step 2 Change inverse trigonometric function to ordinary trigonometric function

Step 3 Draw a right-angle triangle and determine all the three sides based
on trigonometric function obtained in Step 2.

Step 4 Find the relevant trigonometric ratios from a right-angle triangle
obtained in Step 3

Step 5 Evaluate the given question (and refer useful formula)

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Angle Sum and Difference Identities
± = ±

± = ∓
±

± = 1 ∓

Note: ± means you can use plus or minus, and the ∓ means to use the opposite sign

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Double-Angle Identities

2 = 2

2 = 2 − 2
2 = 2 2 − 1
2 = 1 − 2 2
2
2 = 1 − 2

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Reciprocal Trigonometric Identities

1 1 1
= = =

1 1 1
= = =

Pythagorean Identities
2 + 2 = 1

1 + 2 = 2 2 + 1 = 2

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Quotient Identities


=


=

Opposite Angle Identities

− = −
− =
− = −

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MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Evaluating Inverse Trigonometric Functions

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Inverse Trigonometric Functions

Inverse Trigonometric Functions

① Introduction (notation, graph and useful formula)
② Evaluating Inverse Trigonometric Functions using Triangle Method
③ Derivatives of Inverse Trigonometric Functions
④ Integration of Inverse Trigonometric Functions

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Inverse Trigonometric Functions sysmybmoblaobolam.cbo.cm

??How to evaluate inverse
Trigonometric Function
using calculator

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Inverse Trigonometric Functions

?? How to evaluate inverse
Reciprocal Trigonometric
Function using calculator

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Inverse Trigonometric Functions

Evaluating Trigonometric Functions and Their Inverses

Example 1
Without using calculator, evaluate :

Solution :

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Inverse Trigonometric Functions

Coming up next…

Examples Involving Evaluating & Simplifying
Composite Inverse Trigonometric Functions

using Triangle Method

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Stay Tune…

MOOC MAT438/ UiTM

Evaluating Composite Inverse Trigonometric Functions
Using Triangle Method

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Inverse Trigonometric Functions

① Introduction (notation, graph and useful formula)
② Evaluating Inverse Trigonometric Functions using Triangle Method
③ Derivatives of Inverse Trigonometric Functions
④ Integration of Inverse Trigonometric Functions

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Evaluating Composite Inverse Trigonometric Functions Using Triangle Method

Example 2
Without using calculator, evaluate :

Solution :

opposite adjacent
hypotenuse hypotenuse

Therefore,

hypotenuse

adjacent
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Inverse Trigonometric Functions Step 3 Draw a right-angle triangle and determine
all the three sides based on trigonometric
Example 3 function obtained in Step 2.
without using calculator, evaluate

Solution :

Step 1 Let A = inverse trigonometric function

Step 2 Change inverse trigonometric function
to ordinary trigonometric function

adjacent
hypotenuset

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Inverse Trigonometric Functions

Example 3 (continue…)
Without using calculator, evaluate

Solution :

Step 3 Draw a right-angle triangle and determine Step 4 Find the relevant trigonometric ratios
all the sides based on trigonometric from a right-angle triangle obtained in
function obtained in Step 2. Step 3

opposite opposite
adjacent

adjacent Step 5 Evaluate the given question

Therefore,

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Inverse Trigonometric Functions

TUTORIAL 1.1

Try This !

Without using calculator, evaluate

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Inverse Trigonometric Functions

Coming up next…

More Examples Involving Evaluating & Simplifying
Composite Inverse Trigonometric Functions
using Triangle Method

Thank You for Watching…

Stay Tune…

MOOC MAT438/ UiTM

Evaluating Composite Inverse Trigonometric Functions
Using Triangle Method

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Inverse Trigonometric Functions

① Introduction (notation, graph and useful formula)
② Evaluating Inverse Trigonometric Functions using Triangle Method
③ Derivatives of Inverse Trigonometric Functions
④ Integration of Inverse Trigonometric Functions

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Example 4

Solution : Step 3 Draw a right-angle triangle and determine
all the three sides based on trigonometric
Step 1 Let A = inverse trigonometric function function obtained in Step 2.

Step 2 Change inverse trigonometric function
to ordinary trigonometric function

opposite
adjacent

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Example 4 (continue…)

Solution :

Step 3 Draw a right-angle triangle and determine Step 4 Find the relevant trigonometric ratios
all the sides based on trigonometric from a right-angle triangle obtained in
function obtained in. Step 2 Step 3

opposite
hypotenuse

adjacent
hypotenuse

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Example 4 (continue…)
Solution :

Step 5 Evaluate the given question

Therefore,

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Coming up next…

Examples Involving Evaluating & Simplifying
Composite Inverse Trigonometric Functions

using Triangle Method

Thank You for Watching…

Stay Tune…

MOOC MAT438/ UiTM

Evaluating Composite Inverse Trigonometric Functions
Using Triangle Method

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Inverse Trigonometric Functions

① Introduction (notation, graph and useful formula)
② Evaluating Inverse Trigonometric Functions using Triangle Method
③ Derivatives of Inverse Trigonometric Functions
④ Integration of Inverse Trigonometric Functions

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Example 5

Solution : Step 3 Draw a right-angle triangle and determine
all the three sides based on trigonometric
Step 1 Let A = inverse trigonometric function function obtained in Step 2.

Step 2 Change inverse trigonometric function
to ordinary trigonometric function

opposite
adjacent

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Example 5 (continue…)

Solution :

Step 3 Draw a right-angle triangle and determine Step 4 Find the relevant trigonometric ratios
all the sides based on trigonometric from a right-angle triangle obtained in
function obtained in Step 2. Step 3

hypotenuse opposite
hypotenuse

opposite

adjacent
hypotenuse

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Example 5 (continue…)

Solution :

Step 5 Evaluate the given question

Therefore,

01

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Coming up next…

Examples Involving Evaluating & Simplifying
Composite Inverse Trigonometric Functions

using Triangle Method

Thank You for Watching…

Stay Tune…

MOOC MAT438/ UiTM

Evaluating Composite Inverse Trigonometric Functions
Using Triangle Method

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Inverse Trigonometric Functions

① Introduction (notation, graph and useful formula)
② Evaluating Inverse Trigonometric Functions using Triangle Method
③ Derivatives of Inverse Trigonometric Functions
④ Integration of Inverse Trigonometric Functions

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Example 6/ MAR 2014/ MAT238/ Q1a (6 marks)

Solution :

Step 1 Let A = 1st inverse trigonometric function Let B = 2nd inverse trigonometric function

Step 2 Change inverse trigonometric function Change inverse trigonometric function
to ordinary trigonometric function to ordinary trigonometric function

opposite opposite
hypotenuse adjacent

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Example 6/ MAR 2014/ MAT238/ Q1a (6 marks)

Solution :

Step 3 Draw a right-angle triangle and determine Draw a right-angle triangle and determine

all the three sides based on trigonometric all the three sides based on trigonometric

function obtained in Step 2. function obtained in Step 2. opposite

opposite adjacent

hypetonuse

MOOC MAT438/ UiTM

Inverse Trigonometric Functions

Example 6/ MAR 2014/ MAT238/ Q1a (6 marks)

Solution : Step 3

Step 3

Step 4 Find the relevant trigonometric ratios Find the relevant trigonometric ratios
from a right-angle triangle obtained in Step 4 from a right-angle triangle obtained in
Step 3
Step 3

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Inverse Trigonometric Functions

Example 6 (continue…)

Solution : Step 4

Step 4

Step 5 Evaluate the given question

Therefore,

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Inverse Trigonometric Functions Simplify and express in terms of x.

TUTORIAL 1.1b

Try This !

Without using calculator, evaluate

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Inverse Trigonometric Functions

Coming up next…

Derivatives of Inverse Trigonometric Functions

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Stay Tune…

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