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In this article, we have covered various topics for Graphs of Inverse Trigonometric Functions. You can learn about Trigonometric and Inverse Trigonometric Functions which are discussed in the following article. There are various Properties and Formulas of Inverse Trigonometric functions which are discussed in the article.
Read Also: Inverse Trigonometric Functions
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Key Terms: Inverse Trigonometric Functions, Sine curves, Arcus functions, Trigonometric functions, Amplitude, Electronic device, Oscilloscope, Inverse function
Trigonometric Functions
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Trigonometric functions are used to find the measurements like heights of mountains and tall buildings without using measurement tools. Similarly, their inverse functions are widely used in Engineering and other sciences including Physics. For example, an oscilloscope is an electronic device that converts electrical signals into graphs like that of sine function. By manipulating the controls, we can change the amplitude, the period and phase shift of sine curves.
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Inverse Trigonometric functions
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If f(x) is a function then the f(x)-1 is the inverse function. The Trigonometric functions are sin, cos, tan, cot, sec and cosec. Just like the mathematical operations addition and subtraction, the inverse of each other is also similar.
x=sin θ
θ-1= sin-1x (Inverse function)
So, a power of -1 is added to the inverse functions. These inverse trigonometric functions are also called as Arcus functions or cyclometric functions or anti-trigonometric functions. These functions can be represented in the form of graphs.
Properties of Inverse Trigonometric functions
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Following are the Properties of Inverse Trigonometric functions:
| Inverse sine function | Inverse cosine function | Inverse Tangent function | Inverse Cosecant function | Inverse secant function | Inverse Cot function |
|---|---|---|---|---|---|
| Domain: [-1,1] | Domain: [-1,1] | Domain: R | Domain: (-∞,-1] U [1,∞) | Domain: (-∞,-1] U [1,∞) | Domain: R |
| Range: [-π/2, π/2] | Range: [0,π] | Range: [-π/2, π/2] | Range: [-π/2, π/2]-{0} | Range: [0, π]-{π/2} | Range: (0, π) |
Formulas of Inverse Trigonometric Functions
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With respect to the domain and range of the Trigonometric functions, there are some important formulas:
- sin(sin-1x) = x if -1≤ x ≤1 and sin(sin-1y)=y if -π/2 ≤ y ≤ π/2.
- cos(cos-1x)=x if -1≤ x ≤1 and cos(cos-1y)=y if 0 ≤ y(arc cos) ≤ π.
- tan(tan-1x)=x if -∞ < x < ∞ and tan(tan-1y)=y if -π/2 ≤ y(arc tan) ≤ π/2.
- cot(cot-1x)=x if -∞ < x < ∞ and cot(cot-1x)=y if 0
- sec(sec-1x)=x if -∞≤x≤-1 or 1≤x≤∞ and sec(sec-1y)=y if 0≤y≤π, y≠π/2.
- cosec(cosec-1x)=x if -∞≤x≤-1 or 1≤x≤∞ and cosec(cosec-1y)=y if -π/2 ≤ y ≤ π/2, y≠0.
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Graphs of Inverse Trigonometric Functions
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At different angles every trigonometric function has a value ranging from -∞ to ∞. Based on the values graphs are drawn for the Trigonometric functions. The graphs vary from function to function and are as follows:
- Arc sine Function: The sine function is not one to one function in the domain -π/2 ≤ y ≤ π/2. But it changes to be a one to one function in the domain [-π/2, π/2]. When the several points of intersection are joined the graph is in the form of a curve with crests and troughs.
- Arc cosine Function: The cosine function is not one to one function. But it changes to be a one-to-one function in the domain [0,π]. When the several values at different angles are joined, the graph obtained is attached.
- Arc tan function: The tangent function is obtained as a one to one function at the domain [-π/2,π/2]. When the several values at different angles are joined, the graph obtained is attached.
- Arc cot function: The cotangent trigonometric function is the reverse of tangent function. When the several values at different angles are joined, the graph obtained is as attached.
- Arc cosec function: The graph drawn for arc cosec function is different. There is no continuous line or intersection. Two types of line are available in two different quadrants.
- Arc sec function: A curve is obtained on the outside portion between -1 and 1 for the inverse secant function. The graph is as follows.
Things to Remember
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Following are some points which should be cleared by the readers:
- If f(x) is a function then the f(x)-1 is the inverse function. The Trigonometric functions are sin, cos, tan, cot, sec and cosec.
| Inverse sine function | Inverse cosine function | Inverse Tangent function | Inverse Cosecant function | Inverse secant function | Inverse Cot function |
|---|---|---|---|---|---|
| Domain: [-1,1] | Domain: [-1,1] | Domain: R | Domain: (-∞,-1] U [1,∞) | Domain: (-∞,-1] U [1,∞) | Domain: R |
| Range: [-π/2, π/2] | Range: [0,π] | Range: [-π/2, π/2] | Range: [-π/2, π/2]-{0} | Range: [0, π]-{π/2} | Range: (0, π) |
- sin(sin-1x) = x if -1≤ x ≤1 and sin(sin-1y)=y if -π/2 ≤ y ≤ π/2.
- cos(cos-1x)=x if -1≤ x ≤1 and cos(cos-1y)=y if 0 ≤ y(arc cos) ≤ π.
- tan(tan-1x)=x if -∞ < x < ∞ and tan(tan-1y)=y if -π/2 ≤ y(arc tan) ≤ π/2.
- cot(cot-1x)=x if -∞ < x < ∞ and cot(cot-1x)=y if 0
- sec(sec-1x)=x if -∞≤x≤-1 or 1≤x≤∞ and sec(sec-1y)=y if 0≤y≤π, y≠π/2.
- cosec(cosec-1x)=x if -∞≤x≤-1 or 1≤x≤∞ and cosec(cosec-1y)=y if -π/2 ≤ y ≤ π/2, y≠0.
Read Also: Trigonometry Important Formulas
Sample Questions
Ques: Find the principal value of the given equation: y = sin-1(1/√2). [1 Mark]
Ans: Given
y = sin-1(1/√2)
sin(y) = (1/√2)
The range of the principal value branch of sin-1(x) is (−π/2, π/2) and sin(π/4) = 1/√2.
Ques: Find the principal value of the given equation: y = cos-1(1). [1 Mark]
Ans: Given
y = cos-1(1)
cos(y) = 1
We know that the range of the principal value branch of cos-1(x) is (0, π) and cos(0) = 1.
So, the principal value of cos-1(1) = 0.
Ques: Find the principal value of the given equation: y = cosec-1(1). [1 Mark]
Ans: Given
y = cosec-1(√2)
cosec(y) = 1
The range of the principal value branch of cosec-1(x) is [-π/2, π/2] – {0} and cosec(π/2) = 1.
So, the principal value of cosec-1(1) = π/2.
Ques: Find the value of cos(sin-1 0.5). [2 Marks]
Ans: The value of the portion in brackets is an angle.
Noting the range for inverse sine function, we get
sin-1(0.5)=π/6
Hence cos(sin-1(0.5))=cos(π/6)=0.8660.
Ques: Find sin−1(sin5π/4). [2 Marks]
Ans: Since 5π/4>π/2
We know that sin5π/4=−1/√2
Thus, sin−1(sin5π/4)=sin−1(−1/√2)
−π/2≤y≤π/2 and siny=−1/√2
That angle is y=−π/4 , since
sin(−π/4) = −sin(π/4) = −1/√2
Thus, sin−1(sin5π/4)=−π/4
Ques: Find the principal value of the given equation: y = sec-1(1). [2 Marks]
Ans: Given
y = sec-1(1)
sec(y) = 1
Range of the principal value branch of sec-1(x) is [0, π] – {π/2}
sec(0) = 1.
So, the principal value of sec-1(1) = 0.
Ques: Evaluate cos(tan-1(-1)). [1 Mark]
Ans: tan-1(-1)=-π/4
cos(-π/4)=1/2 × √2
Ques: Find the value of y = sec-1(√2). [2 Marks]
Ans: Given
y = sec-1(√2)
sec(y) = (√2)
The range of the principal value branch of sec-1(x) is [0, π] – {π/2} and sec(π/4) = √2.
So, the principal value of sec-1(√2) = π/4.
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