Understanding the Value of Cos 90 and Tan 90
The values of cosine and tangent at 90 degrees (or π/2 radians) are fundamental concepts in trigonometry. These values have specific significance and are often used in various mathematical and scientific applications.
Cos 90
The value of cosine at 90 degrees (or (frac{pi}{2}) radians) is 0. This can be easily understood using the unit circle, which is a circle with a radius of 1 that is centered at the origin of a coordinate system. On this circle, the cosine of an angle represents the x-coordinate of the point where the terminal side of the angle intersects the circle.
At 90 degrees, the point on the unit circle is at (0,1). Therefore, the cosine of 90 degrees is the x-coordinate of this point, which is 0. Mathematically, this is expressed as:
cos 90deg; cos frac{pi}{2} 0
Tan 90
The value of tangent at 90 degrees (or (frac{pi}{2}) radians) is undefined. This is because tangent is defined as the ratio of sine to cosine (tan sin/cos). At 90 degrees, the sine value is 1, while the cosine value is 0. Thus, the expression becomes 1/0, which is mathematically undefined.
Algebraically, we can express this as:
tan 90deg; frac{sin 90deg;}{cos 90deg;} frac{1}{0}
Therefore, tan 90 degrees is undefined.
Using the Unit Circle to Understand Trigonometric Values
The unit circle helps visualize trigonometric values. The x-coordinate of a point on the unit circle represents the cosine of the angle, while the y-coordinate represents the sine of the angle. At 90 degrees, the point on the unit circle is (0, 1).
Here's a diagram to help you visualize the unit circle:
As you can see, at 90 degrees, the x-coordinate is 0, which confirms that cos 90deg; 0.
Conclusion
Understanding the values of cos 90 and tan 90 is crucial for grasping trigonometric concepts. The value of cos 90 is 0, while tan 90 is undefined due to division by zero. These values are foundational to numerous applications in mathematics, engineering, and physics.