Determining the Exact Value of cos 75 Degrees Using Trigonometric Identities

Understanding trigonometric functions has numerous applications in both pure and applied mathematics. One interesting case is finding the exact value of cos 75 degrees. This article explores various methods to calculate the exact value of cos 75 degrees, which is important in many areas, including geometry, physics, and engineering.

Introduction to cos 75 Degrees

The exact value of cos 75 degrees can be calculated using known trigonometric identities. In decimal form, cos 75 degrees is approximately 0.2588190. Additionally, the value can be expressed as a fraction: √6 - √2/4.

Calculating cos 75 Degrees Using Known Values

One common method to find cos 75 degrees is by breaking it down into the sum of two angles, 45 degrees and 30 degrees, and applying the cosine formula:

Method 1: Using Cosine Addition Formula

The cosine of a sum of two angles is given by the formula:

[cos(A B) cos A cos B - sin A sin B]

Substituting A 45° and B 30°, we get:

[cos(75°) cos(45° 30°) cos 45° cos 30° - sin 45° sin 30°]

Using the known values of trigonometric functions:

[cos 45° frac{1}{sqrt{2}}, quad cos 30° frac{sqrt{3}}{2}, quad sin 45° frac{1}{sqrt{2}}, quad sin 30° frac{1}{2}]

The expression becomes:

[cos(75°) frac{1}{sqrt{2}} times frac{sqrt{3}}{2} - frac{1}{sqrt{2}} times frac{1}{2} frac{sqrt{3}}{2sqrt{2}} - frac{1}{2sqrt{2}} frac{sqrt{3} - 1}{2sqrt{2}}]

Therefore, the exact value of cos 75 degrees is:

[cos 75° frac{sqrt{3} - 1}{2sqrt{2}}]

Method 2: Using Alternative Formulas

An alternative method involves expressing cos 75 degrees as a combination of other known values. For instance:

[cos 75° cos (45° 30°)]

Using the cosine addition formula again:

[cos(45° 30°) cos 45° cos 30° - sin 45° sin 30°]

Substituting the known values:

[cos(45° 30°) frac{1}{sqrt{2}} times frac{sqrt{3}}{2} - frac{1}{sqrt{2}} times frac{1}{2} frac{sqrt{3}}{2sqrt{2}} - frac{1}{2sqrt{2}} frac{sqrt{3} - 1}{2sqrt{2}}]

This approach reaffirms the previous result.

Additional Insights on cos 75 Degrees

The value of cos 75 degrees can also be calculated using radians. Since 75 degrees is equivalent to 5π/12 radians, the exact value in radians is:

[cos(5π/12) frac{sqrt{6} - sqrt{2}}{4}]

This can be further simplified to:

[cos 75° frac{sqrt{6} - sqrt{2}}{4} approx 0.2588190]

It is also useful to know that the cosine function is even, meaning:

[cos(-θ) cos(θ)]

Hence:

[cos(-75°) cos 75° frac{sqrt{6} - sqrt{2}}{4} approx 0.2588190]

Conclusion

In conclusion, the exact value of cos 75 degrees is obtained through the use of trigonometric identities and the addition formula. This value, expressed as √6 - √2/4, is crucial in solving various problems in trigonometry and related fields. Understanding these identities and methods enhances one's ability to solve complex trigonometric equations efficiently.

References

1. Trigonometry Identities. (2023). Math Is Fun

2. Trigonometric Functions. (2023). Khan Academy