Why Cos 90 Degrees Equals Zero Using Unit Circle Proof
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 in Trigonometry
1. What is cos 90 degrees equal to?
The value of cos 90° is 0. In trigonometry, cosine represents the ratio of the adjacent side to the hypotenuse in a right triangle. At 90 degrees, the adjacent side becomes zero, so:
- cos 90° = 0
2. Why is cos 90 degrees equal to zero?
Cos 90° equals 0 because at 90°, the adjacent side of a right triangle becomes zero. Using the unit circle definition:
- Cosine is the x-coordinate of a point on the unit circle.
- At 90° (π/2 radians), the point is (0, 1).
- Therefore, cos 90° = 0.
3. What is cos 90 degrees in radians?
Cos 90 degrees in radians is cos(π/2) = 0. Since 90° equals π/2 radians, converting degrees to radians gives:
- 90° × (π/180) = π/2
- Therefore, cos(π/2) = 0
4. How do you find cos 90 degrees using the unit circle?
You find cos 90° using the unit circle by identifying the x-coordinate at 90°. Follow these steps:
- Draw the unit circle with radius 1.
- Locate 90° (π/2) on the circle.
- The coordinate at this angle is (0, 1).
- The cosine value is the x-coordinate.
5. What is the exact value of cos 90 degrees?
The exact value of cos 90° is 0. Unlike other trigonometric ratios that may involve square roots or fractions, cosine at 90 degrees simplifies exactly to zero. This is a fundamental standard angle value in trigonometry.
6. Is cos 90 degrees undefined?
No, cos 90° is not undefined; it equals 0. Cosine is undefined only when dividing by zero in certain contexts, but at 90°, the cosine function has a defined value. In contrast, tan 90° is undefined, not cosine.
7. What is the difference between cos 90° and sin 90°?
The difference is that cos 90° = 0 while sin 90° = 1. On the unit circle at 90°:
- The coordinate is (0, 1).
- Cosine is the x-coordinate = 0.
- Sine is the y-coordinate = 1.
8. How is cos 90 degrees used in trigonometric identities?
Cos 90° = 0 is used to simplify many trigonometric identities and equations. For example:
- In the identity sin²θ + cos²θ = 1, if θ = 90°, then:
- sin²90° + cos²90° = 1² + 0² = 1
9. What is the value of cos 90 degrees in a right triangle?
In a right triangle, cos 90° = 0 because the adjacent side becomes zero at a 90° angle. Since cosine is defined as:
- cos θ = adjacent / hypotenuse
10. What are common mistakes when evaluating cos 90 degrees?
A common mistake is confusing cosine with tangent or sine at 90°. Key points to remember:
- cos 90° = 0
- sin 90° = 1
- tan 90° is undefined
