Inverse Trigonometric Functions — Class 12 Maths Notes

Topics covered in Inverse Trigonometric Functions

1. 1.Introduction of Inverse Trigonometric Functions

   Let's start with something you see almost every day - your smartphone. Imagine you're playing a game on it where you need to slingshot a bird at just the right angle to knock over some blocks. To win, you often adjust the angle of the slingshot. This adjustment and finding out the angle is closely related to trigonometry, and specifically, to what we call inverse trigonometric functions.

   Trigonometric functions (like sine, cosine, and tangent) help us relate the angles of a triangle to the lengths of its sides. But sometimes, we know the sides and need to find the angles. That's where inverse trigonometric functions come in handy. They help us work backward from the sides of a triangle to the angles.

   There are six main inverse trigonometric functions:

   • Arcsine (sin⁻¹)
   • Arccosine (cos⁻¹)
   • Arctangent (tan⁻¹)
   • Arccotangent (cot⁻¹)
   • Arcsecant (sec⁻¹)
   • Arccosecant (csc⁻¹)

   These functions are called "inverse" because they help us find the angle when we know the ratio of the sides. It's like solving a mystery in reverse: instead of finding out what's happening based on the clues (angles), you're figuring out what the clues must have been based on what happened (side lengths).

   Real-Life Application

   Think about architects and engineers. They use inverse trigonometric functions to design buildings, bridges, and roads. When they know the height of a building and the distance from a certain point, they can use these functions to calculate the angles involved in constructing the building.

   How It Works in Math

   Let’s take the inverse sine function (sin⁻¹) as an example. If the sine of an angle θ is 0.5, using sin⁻¹(0.5), we can find out that θ is 30°. This is because sin(30°) = 0.5, and so the arcsine or inverse sine of 0.5 is 30°.

   Activity for You

   Try finding the angle for which the cosine equals 0.5. You'll use the arccosine function for this. It's a simple way to see inverse trigonometric functions in action.

   Careers Using Inverse Trigonometric Functions

   • Engineering: Calculating angles for designs.
   • Architecture: Designing buildings with specific angle requirements.
   • Game Development: Programming angles for movements and trajectories.
   • Navigation: Calculating routes and directions based on coordinates.

   Inverse trigonometric functions are essential tools in various fields, helping professionals solve problems and create solutions that shape our world.

2. 2.Basic Concepts of Inverse Trigonometric Functions

   Inverse trigonometric functions are fascinating because they allow us to find the angles of a triangle when we already know the ratios of its sides. This is incredibly useful in many areas of both everyday life and specialized fields. Let’s break down some basic concepts to understand these functions better.

   1. Definition

   Inverse trigonometric functions are the inverses of the trigonometric functions like sine, cosine, and tangent. These functions are usually denoted as sin⁻¹, cos⁻¹, and tan⁻¹, respectively. They help us determine the angle that corresponds to a given trigonometric ratio.

   2. Domain and Range

   Each inverse trigonometric function has a specific domain (the set of input values it can accept) and range (the set of output values it can produce):

   • sin⁻¹(x) is defined for -1 ≤ x ≤ 1, and its output ranges from -π/2 to π/2 radians.
   • cos⁻¹(x) is also defined for -1 ≤ x ≤ 1, but its output ranges from 0 to π radians.
   • tan⁻¹(x) can take any real number as an input, and its output ranges from -π/2 to π/2 radians (excluding -π/2 and π/2).

   3. Function Behavior

   These functions are also non-linear, meaning they do not form straight lines when graphed. They have specific shapes that are consistent because they are directly related to their corresponding trigonometric functions.

   4. Applications

   We use inverse trigonometric functions in various real-world applications. For example:

   • In navigation, to find the angle of the ship’s direction.
   • In physics, to calculate angles in projectile motion.
   • In computer graphics, to rotate and manipulate images.

   5. Principal Values

   The output of inverse trigonometric functions is called the principal value. For these functions to work properly and to avoid ambiguity (since many angles can have the same sine, cosine, or tangent value), we restrict their outputs to their principal values.

   Example in Daily Life

   Imagine you’re trying to figure out how high a kite is flying. You measure the length of the string fully extended to the kite and the angle it makes with the ground. Using the tangent function and then the inverse tangent function, you can calculate the height of the kite above the ground.

   Careers and Industries

   Knowing these concepts can be incredibly beneficial in careers like:

   • Civil Engineering: Calculating the slopes and angles of roads and bridges.
   • Aerospace Engineering: Designing flight paths and understanding aircraft orientation.
   • Video Game Design: Implementing realistic movements and camera angles.

   Understanding inverse trigonometric functions opens up a world of possibilities for solving problems and designing systems in both your daily life and future career.

3. 3.Find the principal values of the following:

   1. 1. sin−1(−21​)
   2. cos⁡−1(32)2. cos−1(23​​)
   3. csc⁡−1(2)3. csc−1(2)
   4. tan⁡−1(−3)4. tan−1(−3​)
   5. cos⁡−1(−12)5. cos−1(−21​)
   6. tan⁡−1(−1)6. tan−1(−1)

   Solutions:

   1. sin⁡−1(−12)1. sin−1(−21​) The value of −12−21​ is the sine of an angle in the fourth quadrant, since sine is negative there. The reference angle for this in the first quadrant is 30° or 66π​ radians. Therefore, the principal value in radians is −6−6π​.

   2. cos⁡−1(32)2. cos−1(23​​) The value of 3223​​ is the cosine of an angle in the first quadrant, where cosine is positive. This is the cosine of 30° or 66π​ radians. So, the principal value is 66π​ radians.

   3. csc⁡−1(2)3. csc−1(2) The cosecant function is the reciprocal of the sine function. Since sin⁡(30°)=12sin(30°)=21​, then csc⁡−1(2)csc−1(2) corresponds to an angle whose sine is 1221​, which is 30° or 66π​ radians in the first quadrant. Hence, the principal value is 66π​ radians.

   4. tan⁡−1(−3)4. tan−1(−3​) The value −3−3​ is the tangent of an angle in the fourth quadrant, as tangent is negative there. This corresponds to the tangent of 60° or 33π​ radians, but since it's negative, we take the negative of this angle. So, the principal value is −3−3π​ radians.

   5. cos⁡−1(−12)5. cos−1(−21​) The value −12−21​ is the cosine of an angle in the second quadrant, where cosine is negative. The reference angle where cosine is 1221​ is 60° or 33π​ radians, so the principal value is −3=23π−3π​=32π​ radians.

   6. tan⁡−1(−1)6. tan−1(−1) The value −1−1 is the tangent of an angle in the fourth quadrant. The reference angle where the tangent is 1 is 45° or 44π​ radians. Since we are looking for a negative angle, the principal value is −4−4π​ radians.

4. 4.Find the values of the following:

   Question 1:

   tan⁡−1(1)+cos⁡−1(−12)+sin⁡−1(−12)tan−1(1)+cos−1(−21​)+sin−1(−21​)

   Solution 1:

   • tan⁡−1(1)tan−1(1) is the angle whose tangent is 1. This is 44π​ radians or 45°.
   • cos⁡−1(−12)cos−1(−21​) is the angle whose cosine is -1/2. This occurs at 2332π​ radians or 120°.
   • sin⁡−1(−12)sin−1(−21​) is the angle whose sine is -1/2. This occurs at −6−6π​ radians or -30°.

   Adding these up: 4+23−6=312+812−212=912=344π​+32π​−6π​=123π​+128π​−122π​=129π​=43π​ radians.

   Question 2:

   cos⁡−1(12)+2sin⁡−1(12)cos−1(21​)+2sin−1(21​)

   Solution 2:

   • cos⁡−1(12)cos−1(21​) is the angle whose cosine is 1/2. This is 33π​ radians or 60°.
   • sin⁡−1(12)sin−1(21​) is the angle whose sine is 1/2. This is 66π​ radians or 30°. Multiplying by 2 gives 33π​ radians.

   Adding these up: 3+3=233π​+3π​=32π​ radians.

   Question 3:

   If sin⁡−1=sin−1(x)=y, then:

   Solution 3:

   The range of the principal value of the sine inverse function (sin⁡−1sin−1(x)) is −2−2π​ to 22π​. So, y should lie within this range. Thus, the correct option is:

   (B) −2≤≤2−2π​≤y≤2π​

   Question 4:

   tan⁡−1(−3)−sec⁡−1(−2)tan−1(−3​)−sec−1(−2) is equal to:

   Solution 4:

   • tan⁡−1(−3)tan−1(−3​) is the angle whose tangent is −3−3​. This is −3−3π​ radians or -60°.
   • sec⁡−1(−2)sec−1(−2) is the angle whose secant is -2. Since secant is the reciprocal of cosine, we are looking for an angle whose cosine is -1/2 in the second quadrant, which is 2332π​ radians or 120°.

   Subtracting these: −3−23=−−3π​−32π​=−π radians.

   The negative sign indicates the direction of the angle, but since we are usually looking for a positive principal value, we take the equivalent positive angle which is 2−=2π−π=π radians.

   Therefore, the answer to Question 14 is: (A) π

5. 5.Properties of Inverse Trigonometric Functions

   Understanding Inverse Functions:

   First, you need to understand that an inverse function basically reverses the action of the original function. For trigonometric functions, which usually take an angle and give you a ratio of sides of a triangle, their inverses take a ratio and give you an angle.

   Principal Values:

   Since trigonometric functions are periodic (they repeat their values over intervals), without restriction, their inverses would give us many angles for a single ratio. However, we only want one angle, the most commonly used one, which we call the principal value. For each inverse trigonometric function, we restrict the output to a certain interval:

   • For arcsine (sin⁡−1sin−1), this interval is [−2,2][−2π​,2π​].
   • For arccosine (cos⁡−1cos−1), the interval is [0,][0,π].
   • For arctangent (tan⁡−1tan−1), the interval is (−2,2)(−2π​,2π​).

   The Property of Inverses:

   The fundamental property of an inverse function says that the function followed by its inverse gets you back to where you started. In the case of trigonometric functions:

   • sin⁡(sin⁡−1)=sin(sin−1(x))=x, for all x in the interval [−1,1][−1,1] because the sine of an angle is always between -1 and 1.
   • sin⁡−1(sin⁡)=sin−1(sin(y))=y, for all y in the interval [−2,2][−2π​,2π​] because we've restricted arcsine to this interval to get principal values.

   This means, if you have a ratio (like 1/2), and you apply arcsine to it, you'll get an angle. If you then take the sine of that angle, you'll get back the same ratio (1/2).

   Why These Properties Matter:

   This concept is really important when solving equations involving trigonometric functions, because you can switch between angles and ratios knowing that you'll always get the right 'principal' angle out, as long as you're within the correct interval.

   Examples of Applying the Property:

   1. If you have the equation sin⁡=1/2sin(y)=1/2, you can solve for y by taking the arcsine of both sides to get =sin⁡−1(1/2)y=sin−1(1/2). The value of y will be /6π/6 because that’s the angle whose sine is 1/2, and it’s within the principal value range of arcsine.

   2. If an equation says cos⁡=−1/2cos(y)=−1/2, and you want to solve for y, you'd use arccosine: =cos⁡−1(−1/2)y=cos−1(−1/2). The solution will be 2/32π/3 because that’s the angle whose cosine is -1/2, and it falls within the principal value range of arccosine.

   Limitations:

   These properties are powerful, but they come with a caveat—they only work within those specific intervals (the 'principal value' ranges). If you use angles outside these ranges, you may not get back the original value because you'll be outside the 'circle of trust', and the function won't behave as expected.

   Conclusion:

   Understanding these properties lets you move confidently between angles and ratios, which is a fundamental skill in trigonometry. This is essential in fields like physics (for example, analyzing waves), engineering (like determining the angles within structures), and even in things like art (creating patterns with specific angles) and navigation (calculating routes based on compass bearings).

   Keeping It Real:

   In real life, these functions help with a lot. Pilots use them to fly planes at the correct angles, engineers use them to make sure bridges are angled correctly, and even in your smartphone for tilt controls in video games.

   So, inverse trigonometric functions are the detectives of the math world. They take the clues (ratios) and solve the mystery (find the angle).

   And remember, these detectives only work with certain clues (values), and they give you the best answer (principal value) to your angle question!

6. 6.Write the following functions in the simplest form

   1. tan⁡−1(1+2−1)1. tan−1(x1+x2​−1​), ≠0x=0

   To simplify this, we can use the following identity: tan⁡(2)=1+sin⁡−1cos⁡/sin⁡tan(2θ​)=cos(θ)/sin(θ)1+sin(θ)​−1​ Comparing this with our expression, we can deduce that: 1+21+x2​ represents 1+sin⁡21+sin2(θ)​ and x represents sin⁡cos⁡cos(θ)sin(θ)​. Thus, our expression represents tan⁡(2)tan(2θ​), and the simplest form of this function would be 22θ​.

   1. tan⁡−1(1−cos⁡1+cos⁡)2. tan−1(1+cos(x)​1−cos(x)​), 0<<0<x<π

   Here we can use the half-angle identities: sin⁡2(2)=1−cos⁡2sin2(2x​)=21−cos(x)​ cos⁡2(2)=1+cos⁡2cos2(2x​)=21+cos(x)​

   So the expression becomes: tan⁡−1(2sin⁡(2)2cos⁡(2))=tan⁡−1(tan⁡(2))tan−1(2​cos(2x​)2​sin(2x​)​)=tan−1(tan(2x​))

   Thus, the simplest form is 22x​.

   1. tan⁡−1(cos⁡−sin⁡cos⁡+sin⁡)3. tan−1(cos(x)+sin(x)cos(x)−sin(x)​), 4<<344π​<x<43π​

   This expression can be simplified by dividing the numerator and the denominator by cos⁡cos(x), which gives us: tan⁡−1(1−tan⁡1+tan⁡)tan−1(1+tan(x)1−tan(x)​)

   Using the tangent subtraction formula: tan⁡(−)=tan⁡−tan⁡1+tan⁡tan⁡tan(A−B)=1+tan(A)tan(B)tan(A)−tan(B)​

   We can see that our expression represents tan⁡−1(tan⁡(−/4))tan−1(tan(x−π/4)), so the simplest form is −4x−4π​.

   1. tan⁡−1(2−2)4. tan−1(a2−x2​x​), ∣∣<∣x∣<a

   This is directly the inverse tangent function applied to a ratio of a right-angled triangle with opposite side x and adjacent side 2−2a2−x2​. This forms the angle whose tangent is /2−2x/a2−x2​. So, it's already in the simplest form.

   1. tan⁡−1(32−33−32)5. tan−1(a3−3ax23a2x−x3​), >0;−3<<3a>0;−3​a​<x<3​a​

   This looks like the tangent addition formula: tan⁡(+)=tan⁡+tan⁡1−tan⁡tan⁡tan(A+B)=1−tan(A)tan(B)tan(A)+tan(B)​ Where ==tan⁡−1(/)A=B=tan−1(x/a), because tan⁡(tan⁡−1(/))=/tan(tan−1(x/a))=x/a. Therefore, the simplest form is 2tan⁡−1(/)2tan−1(x/a), because adding the angle to itself is doubling it.

   These simplified expressions are derived from well-known trigonometric identities and the properties of inverse trigonometric functions. They demonstrate the elegance of inverse trigonometry in reducing complex-looking expressions to simpler forms.

7. 7.Find the values of each of the following:

   Question 1:

   Find the value of tan⁡−1[2cos⁡(2sin⁡−112)]tan−1[2cos(2sin−121​)].

   Solution 1:

   First, we find 2sin⁡−1122sin−121​. Since sin⁡−112sin−121​ is 66π​ or 30 degrees (because sine of 30 degrees is 1/2), we multiply this by 2 to get 33π​ or 60 degrees.

   Now, cos⁡(3)cos(3π​) is 1221​. Therefore, 2cos⁡(3)2cos(3π​) is 11.

   Thus, tan⁡−1(1)tan−1(1) is 44π​ or 45 degrees, because the tangent of 45 degrees is 1.

   Question 2:

   Find the value of 12tan⁡−1(−21+2)+12cos⁡−1(−21+2)21​tan−1(1+x2−2x​)+21​cos−1(1+y2−y2​), given ∣∣<1∣x∣<1, >0y>0, and <1y<1.

   Solution 2:

   Both parts of this expression resemble the formulas for tangent and cosine of double angles.

   For the first part, the double angle formula for tangent is tan⁡(2)=2tan⁡1−tan⁡2tan(2A)=1−tan2(A)2tan(A)​. For our case, tan⁡=−tan(A)=−x, so tan⁡−1(−21+2)tan−1(1+x2−2x​) simplifies to −2−2A, where =tan⁡−1A=tan−1(x). So the first half of the expression becomes 12×−2=−21​×−2A=−A, which is −tan⁡−1−tan−1(x).

   For the second part, using the double angle formula for cosine, cos⁡(2)=1−21+2cos(2B)=1+y21−y2​. For our case, if we take B such that cos⁡=cos(B)=y (since y is positive and less than 1), then cos⁡−1(−21+2)cos−1(1+y2−y2​) simplifies to −2π−2B, where =cos⁡−1B=cos−1(y). So the second half of the expression becomes 12×(−2)=2−21​×(π−2B)=2π​−B, which is 2−cos⁡−12π​−cos−1(y).

   Combining both parts, we get: −tan⁡−1+2−cos⁡−1−tan−1(x)+2π​−cos−1(y)

8. 8.A Table for Domains and Ranges (principal value branches) of inverse trigonometric functions are given in the following table:

   Below is a table that details the domains and ranges (also known as the principal value branches) for the basic inverse trigonometric functions:

   Function	Domain of Inverse Function	Range of Inverse Function (Principal Value Branch)
   sin⁡−1sin−1(x)	[−1,1][−1,1]	[−2,2][−2π​,2π​]
   cos⁡−1cos−1(x)	[−1,1][−1,1]	[0,][0,π]
   tan⁡−1tan−1(x)	(−∞,∞)(−∞,∞)	(−2,2)(−2π​,2π​)
   cot⁡−1cot−1(x)	(−∞,∞)(−∞,∞)	(0,)(0,π)
   sec⁡−1sec−1(x)	[−∞,−1]∪[1,∞)[−∞,−1]∪[1,∞)	[0,2)∪(2,][0,2π​)∪(2π​,π]
   csc⁡−1csc−1(x)	[−∞,−1]∪[1,∞)[−∞,−1]∪[1,∞)	[−2,0)∪(0,2][−2π​,0)∪(0,2π​]

   This table summarizes the allowed inputs (domains) and the corresponding outputs (ranges) for each inverse trigonometric function. These ranges are the principal values where the inverse functions are generally defined.