Tan Pi 4

David Muir

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08-12-2024

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The expression "tan π/4" refers to the tangent of the angle π/4 radians, which is equivalent to 45 degrees. In mathematics, the tangent function (tan) is defined as the ratio of the sine to the cosine of an angle. Therefore, to calculate tan π/4, we need to determine the values of sin(π/4) and cos(π/4).

Calculating sin(π/4) and cos(π/4)

The sine and cosine of π/4 can be derived from the unit circle or a 45-45-90 right-angled triangle. In a 45-45-90 triangle, the two legs are equal in length, and the hypotenuse is √2 times the length of a leg. If we consider a unit circle (a circle with radius 1), a 45-degree angle intersects the circle at the point (√2/2, √2/2). The x-coordinate represents the cosine, and the y-coordinate represents the sine.

Therefore:

• sin(π/4) = √2/2
• cos(π/4) = √2/2

Calculating tan(π/4)

Now, we can calculate tan(π/4) using the definition:

tan(π/4) = sin(π/4) / cos(π/4) = (√2/2) / (√2/2) = 1

Therefore, the tangent of π/4 is equal to 1.

Significance and Applications

The value of tan(π/4) = 1 has several significant applications in various fields, including:

• Trigonometry: It is a fundamental value used in solving trigonometric equations and identities.
• Calculus: It appears in derivatives and integrals of trigonometric functions.
• Physics and Engineering: It's crucial in calculations involving angles, vectors, and oscillations.

In conclusion, understanding the calculation and significance of tan π/4 is essential for anyone working with trigonometry and related fields. Its simple value of 1 belies its importance in numerous mathematical and scientific applications.