How to Calculate Inverse Cosine on Calculator and with Formulas
The concept of Inverse Cosine is a crucial part of trigonometry. It helps us find the angle when a cosine value is given and appears frequently in board exams, JEE, NEET, and real-life applications like physics and engineering.
What Is Inverse Cosine?
Inverse Cosine, commonly written as arccos or cos⁻¹(x), is the function used to find the angle whose cosine value is a specific number. For example, if cos(θ) = x, then θ = cos⁻¹(x). This is not to be confused with the reciprocal of cosine (which is secant or sec). You’ll see inverse cosine used whenever you need to reverse a cosine calculation, such as in the trigonometric functions chapter, in vector direction problems in physics, and also in geometry tasks.
Key Formula for Inverse Cosine
Here’s the standard formula: \( \cos^{-1}(x) = \theta \quad \text{if and only if} \quad \cos(\theta) = x \), where \( x \) is between –1 and 1.
| Notation | Meaning |
|---|---|
| arccos(x) | Inverse cosine of x |
| cos⁻¹(x) | Angle whose cosine is x |
Domain and Range of Inverse Cosine
It’s important to remember that cos⁻¹(x) is defined only for x values between –1 and 1.
| Domain | Range (Radians) | Range (Degrees) |
|---|---|---|
| –1 ≤ x ≤ 1 | 0 ≤ θ ≤ π | 0° ≤ θ ≤ 180° |
How to Calculate Inverse Cosine (cos⁻¹x): Step-by-Step
- Check if the value of x is between –1 and 1.
Example: x = 1/2 (valid), x = 2 (not valid)
- On your calculator, press the ‘SHIFT’ or ‘2nd’ key, then ‘COS’ to access cos⁻¹.
- Enter the value of x and select the desired mode (degrees or radians).
- Press ‘=’ and read the angle. For cos⁻¹(1/2): Most calculators show 60°, which can also be written as π/3 radians.
- Negative values return angles greater than 90° and up to 180°.
Common Values for cos⁻¹(x)
| x | cos⁻¹(x) in Degrees | cos⁻¹(x) in Radians |
|---|---|---|
| 1 | 0° | 0 |
| 1/2 | 60° | π/3 |
| 0 | 90° | π/2 |
| –1/2 | 120° | 2π/3 |
| –1 | 180° | π |
Cross-Disciplinary Usage
Inverse Cosine is not only useful in Maths but also plays a crucial role in Physics (finding angles, resolving forces), Computer Science (graphics/animation), and engineering (signal resolution, navigation). Students preparing for JEE and NEET will encounter problems requiring cos⁻¹ frequently.
Step-by-Step Illustration: Solved Example
Example: Find the angle θ such that cos θ = 0.
1. We need to find θ for which cos θ = 0.2. θ = cos⁻¹(0).
3. From the value table, cos⁻¹(0) = 90° or π/2 radians.
4. Final answer: θ = 90° (or π/2 radians).
Speed Trick or Shortcut
Here’s a trick: For quick calculation of cos⁻¹(special values), remember the standard values (such as 0, ±1/2, ±1). These are often used in MCQs. For cos⁻¹(–1/2), directly recall it is 120° (2π/3 radians) for the principal value.
Tricks like this are practical for exams. Vedantu’s live classes share more such shortcut methods for trigonometric inverses.
Try These Yourself
- Find cos⁻¹(–1).
- Calculate the angle whose cosine is 1/2.
- What happens if you try cos⁻¹(2)?
- Find all angles θ in [0°, 180°] for which cos θ = 0.
Frequent Errors and Misunderstandings
- Confusing inverse cosine (cos⁻¹) with cosine reciprocal (sec). Remember, sec(x) = 1/cos(x), NOT cos⁻¹(x).
- Trying to calculate cos⁻¹(x) for x values outside –1 to 1.
- Forgetting that calculator returns only the principal value (within [0°, 180°]).
Relation to Other Concepts
Inverse Cosine is related closely to sin inverse (arcsin) and tan inverse (arctan). Understanding cos⁻¹(x) will make it easier to deal with trigonometric identities and equations. For standard values, refer to the trigonometry table and trigonometric identities page.
Classroom Tip
To quickly remember the domain and range: The input for cos⁻¹(x) must be between –1 and 1, and the output is always an angle between 0° and 180°. Vedantu’s teachers recommend memorizing the “cosine curve” on the unit circle so you always know which angles to expect.
We explored Inverse Cosine—from its definition, formula, domain/range, key values, mistakes, and close links with other trig functions. Continue learning and practicing with Vedantu for more shortcuts and expert guidance in trigonometry!
For further reading, visit: Trigonometric Functions | Trigonometric Identities | Sin Inverse (arcsin) in Maths | Trigonometry Table | Tan Inverse (arctan) in Maths
FAQs on Inverse Cosine (arccos or cos⁻¹) – Definition, Graph, Formula & Examples
1. What is inverse cosine in Maths?
Inverse cosine, also known as arccos or cos-1, is a trigonometric function that determines the angle whose cosine is a given value. For example, if cos(θ) = x, then cos-1(x) = θ. The function is defined for x values between -1 and 1 (inclusive).
2. How do you find the angle using cos⁻¹?
To find an angle using cos-1, you input the cosine value into the function. Most calculators and software have a dedicated arccos button or function. Ensure your calculator is set to the correct angle mode (degrees or radians) before calculating. For example, to find the angle whose cosine is 0.5, you would calculate cos-1(0.5). In degrees, this is 60°; in radians, it's π/3.
3. Is inverse cosine the same as 1/cos?
No. 1/cos(x) is the secant function (sec(x)), which is the reciprocal of the cosine function. The inverse cosine function, cos-1(x), is entirely different; it finds the angle whose cosine is x, not the reciprocal of the cosine.
4. What is the domain and range of arccos(x)?
The domain of arccos(x) is [-1, 1], meaning you can only input values between -1 and 1 (inclusive). The range of arccos(x) is [0, π] radians or [0°, 180°] degrees. This restricted range ensures that the inverse cosine function is a proper function (one output for each input).
5. How do you solve cos⁻¹(1/2) and cos⁻¹(-1/2)?
To solve cos-1(1/2), you find the angle whose cosine is 1/2. This is 60° (π/3 radians). For cos-1(-1/2), you find the angle whose cosine is -1/2. Since the range of arccos is [0, π], the solution is 120° (2π/3 radians).
6. Why does cos inverse sometimes show “Math Error” on calculators?
A "Math Error" appears if you try to calculate the inverse cosine of a number outside its domain, which is -1 to 1. The cosine function never produces values outside this range, so the inverse cosine is undefined for numbers outside this interval.
7. How do negative values of x affect cos⁻¹(x) output?
Negative values of x in cos-1(x) result in angles in the second quadrant (between 90° and 180° or π/2 and π radians). The output angle will always be between 90° and 180° because of the restricted range of the inverse cosine function.
8. Why isn’t the inverse of cosine the same as secant?
The inverse of a function reverses the input and output. The secant function is the reciprocal of the cosine, not its inverse. Inverse cosine finds the angle, while the secant is a different trigonometric function altogether.
9. What common exam mistakes happen with inverse cosine?
Common mistakes include: forgetting the restricted range of arccos (0 to 180 degrees), confusing it with the secant function, incorrectly using the calculator (radians vs degrees mode), and misinterpreting negative input values.
10. How is inverse cosine used in real life applications?
Inverse cosine has applications in various fields like physics and engineering, particularly where angles need to be determined from known cosine values. Examples include calculating angles in triangle problems, vector analysis, and wave phenomena.
11. What are the properties of the inverse cosine function?
The inverse cosine function is a monotonically decreasing function. Its graph is a reflection of the cosine function over the line y = x, for the restricted domain and range. It's also continuous within its domain.
12. How can I use the arccos function in formulas and equations?
The arccos function is used in formulas where you need to find the angle given its cosine. It is often combined with other trigonometric functions and algebraic operations. The specific application depends on the context of the problem; for instance, you might see arccos in equations related to finding angles in triangles or solving trigonometric equations.
