Why Is Cos 90 Degrees Equal to Zero?
The concept of cos 90 degrees plays a key role in mathematics, especially within trigonometry, and is widely applicable to both real-life situations and competitive exam scenarios. Knowing standard trigonometric values like cos 90° makes solving problems in geometry and physics much faster and easier.
What Is Cos 90 Degrees?
Cos 90 degrees means the value of the cosine trigonometric function for the angle 90°. Cos 90° is used in trigonometric tables and formulas, especially while solving questions in coordinate geometry, triangles, and physics. The cosine of 90 degrees equals 0. This means, when you take the cosine of a right angle, the result is always zero. Cos 90 also appears as cosine 90 degrees in some books, and it’s useful for understanding the unit circle, standard trig tables, and why certain values drop out in formulae.
Key Formula for Cos 90 Degrees
Here’s the standard formula: \( \cos(90^\circ) = 0 \)
You can also write 90 degrees as radians, where \( 90^\circ = \frac{\pi}{2} \) radians. So, \( \cos\left(\frac{\pi}{2}\right) = 0 \).
Trigonometric Table for Standard Angles
| Angle (°) | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| Cosine Value | 1 | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{\sqrt{2}} \) | \( \frac{1}{2} \) | 0 |
Cos 90° Using the Unit Circle
The unit circle helps visualise why cos 90° equals 0. On the unit circle, angles are measured from the positive x-axis. The x-coordinate at 90°, which is the top point on the circle, is 0. That is why the cosine of 90 degrees is zero. This is a widely used proof in all trigonometry textbooks. For more about the unit circle and diagrams, visit Unit Circle on Vedantu.
Step-by-Step Derivation: Why is Cos 90 Degrees Zero?
- Draw a circle with center at (0,0) and a radius of 1 unit (unit circle).
- The 90° angle is at the top of the circle, coordinate (0, 1).
- Cosine of any angle = x-coordinate at that angle.
- So, x at 90° = 0 → cos 90° = 0
Cos 90 Degrees in Formulas and Identities
Cos 90 often appears in trigonometric identities. Examples include:
- In the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
- As a result, \( \sin^2 90^\circ + \cos^2 90^\circ = 1^2 + 0^2 = 1 \)
- In sum and difference formulas like \( \cos(90° - x) = \sin x \)
- In right triangle problems, adjacent side for 90° = 0, so cos 90° is 0.
- As a denominator, cos 90° = 0, so the expression becomes undefined (division by zero).
Real-World Applications
Cos 90 degrees is used in:
- Physics: Resolving force components along axes, where a direction is perpendicular (angle 90°), the component is zero.
- Engineering: Analyzing vector projections—component vanishes when vectors are perpendicular.
- Mathematics: Area of triangles using trigonometry, and transformations on coordinate axes.
- Exam Shortcuts: Recognizing values that drop out (become 0) for pure right angles without calculation.
Common Mistakes with Cos 90 Degrees
- Assuming cos 90° = 1 (it’s 0, while sin 90° = 1).
- Forgetting that cos 90° as denominator gives undefined expressions (division by zero).
- Mixing up degree and radian measure for calculators—always convert 90° to \( \frac{\pi}{2} \) radians, if needed.
Memory Tip: How to Remember Cos 90 Degrees
One easy mnemonic is “C starts at 1, S at 0”—cosine table values march down from 1 to 0 as the angle rises from 0° to 90°. Vedantu teachers often remind students, “At 90°, cosine stands on the y-axis, so there’s no horizontal/adjacent side left—cos 90 is zero!”
Examples Using Cos 90 Degrees
Example 1: Simplify \( 5 \frac{\cos 90^\circ}{\sin 90^\circ} \)
1. \( \cos 90^\circ = 0 \), \( \sin 90^\circ = 1 \ )2. \( 5 \times \frac{0}{1} = 0 \)
Final Answer: 0
Example 2: What is \( 2 \cos(90^\circ) + 3 \cos(0^\circ) \)?
1. \( \cos(90^\circ) = 0 \); \( \cos(0^\circ) = 1 \ )2. \( (2 \times 0) + (3 \times 1) = 3 \)
Final Answer: 3
Try These Yourself
- Write the formula to convert 90 degrees into radians and find cos of the result.
- Use the unit circle to show why cos 90° does not equal 1.
- If \( \sin^2 \theta + \cos^2 \theta = 1 \), fill in the values for θ = 90°.
Relation to Other Concepts
The idea of cos 90 degrees is closely connected with sin 90 degrees, cos 0 value, and the trigonometric values table. Mastering cos 90 is essential for working with trigonometric ratios and advanced topics in coordinate geometry and calculus.
We explored cos 90 degrees—its meaning, value, formula, proofs, examples, and tricks for memory. Keep using Vedantu’s learning resources and live classes to practice such standard values. They are foundational for exam confidence and advanced maths concepts!
Useful Vedantu Links: Sin 90 Degrees | Cos 0 Value | Trigonometric Values | Unit Circle | Trigonometric Ratios
FAQs on Cos 90 Degrees – Value, Proof & Meaning in Trigonometry
1. What is the exact value of Cos 90 degrees?
The exact value of Cos 90 degrees (or Cos 90°) is 0. This is a fundamental value in trigonometry, representing the cosine of a right angle. In the radian system, 90 degrees is equal to π/2, so cos(π/2) also equals 0.
2. Why is Cos 90 degrees equal to 0? How can this be proven?
Cos 90° is zero because at a 90-degree angle, the adjacent side of the angle collapses to zero. There are two simple ways to prove this as per the NCERT syllabus:
- Using a Right-Angled Triangle: In trigonometry, Cos θ = (Adjacent Side) / (Hypotenuse). As the angle θ approaches 90°, the length of the adjacent side approaches 0. Therefore, Cos 90° = 0 / Hypotenuse = 0.
- Using the Unit Circle: On a unit circle, the x-coordinate of any point on the circle represents the cosine of the angle. At 90°, the point lies on the y-axis at (0, 1). Since the x-coordinate is 0, the value of Cos 90° is 0.
3. How is the value of Cos 90° different from Sin 90°?
The primary difference lies in their values and what they represent on the unit circle. While Cos 90° = 0, the value of Sin 90° = 1. On the unit circle, Cos 90° corresponds to the x-coordinate at the 90° mark (which is 0), whereas Sin 90° corresponds to the y-coordinate at the same point (which is 1). This highlights their complementary nature.
4. What is the value of Cos 90 in fractions?
The value of Cos 90 degrees is 0. As a fraction, zero can be written as 0/1. This is because any fraction with 0 in the numerator (and a non-zero denominator) is equal to zero.
5. What does the identity Cos(90° - θ) = Sin θ tell us about the 90-degree angle?
This is a fundamental co-function identity that shows the relationship between sine and cosine. It explains that the cosine of an angle is equal to the sine of its complementary angle (the angle that adds up to 90°). For example, if you set θ = 0°, the identity becomes Cos(90°) = Sin(0°). Since we know Sin 0° is 0, this provides another way to confirm that Cos 90° = 0.
6. What are some real-world applications of Cos 90 degrees?
The concept that Cos 90° = 0 is crucial in various fields where perpendicular relationships are involved. Key applications include:
- Physics: When calculating work done by a force (Work = Force × Distance × Cos θ), if the force is applied perpendicular (at 90°) to the direction of motion, the work done is zero because Cos 90° = 0.
- Engineering and Vectors: When resolving a vector into its components, the component along a perpendicular axis is always zero. This is used extensively in mechanics and structural analysis.
- Navigation: It helps in calculating paths and bearings where directions are perpendicular to each other, ensuring components in certain directions cancel out.
7. What are the most common mistakes to avoid when using Cos 90° in problems?
Students often make a few common errors when dealing with Cos 90°. The key mistakes to avoid are:
- Confusing it with Sin 90°: A frequent error is mixing up the values and assuming Cos 90° = 1. Always remember Cos 90° = 0 and Sin 90° = 1.
- Division by Zero: Forgetting that Cos 90° is zero can lead to errors in expressions like tan 90° (which is sin 90°/cos 90°). Since the denominator becomes 0, the expression is undefined.
- Calculator Mode: Using a calculator set to 'Radians' mode when the input is 'Degrees' will give an incorrect answer. Ensure your calculator is in the correct mode for the problem.
