Understanding Trigonometric Functions at 315°
Trigonometry is a subfield of mathematics that studies the relationships between the angles and sides of triangles. Although it primarily deals with right triangles, its principles apply to any triangle and are extended to the study of angles in a broader sense. Understanding the values of trigonometric functions at specific angles, such as 315°, is crucial for various applications in both pure and applied mathematics.
What is 315° in Trigonometry?
315° is an angle that can be found in the fourth quadrant of the unit circle. Angles in trigonometry are often measured in either degrees or radians. In the unit circle, an angle of 315° is equivalent to 7π/4 radians. This angle is 45° less than a full rotation of 360°, making it easy to deduce its trigonometric values by understanding the reference angles and the quadrants.
Trigonometric Functions at 315°
Let's explore the six primary trigonometric functions at 315°: sine, cosine, tangent, cosecant, secant, and cotangent.
Sine of 315°
The sine of 315° is calculated as:
sin 315° sin (360° - 45°) -sin 45°
Using the value sin 45° √2/2, we get:
sin 315° -√2/2
Cosine of 315°
The cosine of 315° is calculated similarly:
cos 315° cos (360° - 45°) cos 45°
Using the value cos 45° √2/2, we get:
cos 315° √2/2
Tangent of 315°
The tangent function is the ratio of sine to cosine:
tan 315° sin 315° / cos 315°
Using the known values for sine and cosine, we get:
tan 315° -√2/2 / √2/2 -1
Cosecant of 315°
The cosecant of an angle is the reciprocal of the sine:
csc 315° 1 / sin 315° 1 / -√2/2 -√2
Secant of 315°
The secant of an angle is the reciprocal of the cosine:
sec 315° 1 / cos 315° 1 / √2/2 √2
Cotangent of 315°
The cotangent of an angle is the reciprocal of the tangent:
cot 315° 1 / tan 315° 1 / -1 -1
Application of Trigonometric Functions at 315°
Knowing the trigonometric values at 315° has practical applications in various fields, such as physics, engineering, and construction. For instance, in physics, it can help in calculating the components of forces and velocities. In engineering, it is used for analyzing structures and components.
Summary
In summary, the trigonometric functions at 315° are:
Sine (sin 315°): -√2/2 Cosine (cos 315°): √2/2 Tangent (tan 315°): -1 Cosecant (csc 315°): -√2 Secant (sec 315°): √2 Cotangent (cot 315°): -1Understanding these values is essential for further studies in trigonometry and its related fields.