Question

Use the unit circle to evaluate the six trigonometric functions of \(\theta\). \(\theta=540^{\circ}\)

Short answer

The six trigonometric functions of \(540^{\circ}\) are: \(\sin(540^{\circ})=0\), \(\cos(540^{\circ})=-1\), \(\tan(540^{\circ})=0\), \(\csc(540^{\circ})\) is undefined, \(\sec(540^{\circ})=1\), \(\cot(540^{\circ})\) is undefined.

Step-by-step solution

Reduce Angle to within a Full Rotate

First, it's important to reduce an angle that is larger than 360 degrees to between 0 and 360 degrees. A full rotate (360 degree) doesn't change the position of the point on the unit circle. Therefore, subtract 360 from \(\theta\) until you get an angle less than or equal to 360. Here the equivalent angle will be \(540^{\circ} - 360^{\circ} = 180^{\circ}\)

Evaluate Sine, Cosine and Tangent

Then, evaluate sine, cosine and tangent for the equivalent angle. On the unit circle, the sine value corresponds to the y-coordinate, the cosine value to the x-coordinate, and the tangent is the ratio of the y-coordinate (sine) over the x-coordinate (cosine). The equivalent angle 180 degrees corresponds to the point (-1, 0) on the unit circle. So, \(\sin(180^{\circ}) = 0\), \(\cos(180^{\circ}) = -1\), and \(\tan(180^{\circ}) = 0/-1 = 0\) (since 0 divided by any number is 0)

Evaluate Cosecant, Secant and Cotangent

Finally, evaluate the reciprocal of sine, cosine and tangent, which are cosecant, secant, and cotangent respectively. The cosecant is the reciprocal of sine, the secant is the reciprocal of cosine, and the cotangent is the reciprocal of tangent. Since \(\sin(180^{\circ}) = 0\), \(\csc(180^{\circ})\) is undefined because we cannot divide by zero. For \(\cos(180^{\circ}) = -1\), \(\sec(180^{\circ}) = -1/-1 = 1\). As \(\tan(180^{\circ}) = 0\), \(\cot(180^{\circ})\) is also undefined because we cannot divide by zero.

Key concepts

Trigonometric Functions

Understanding trigonometric functions is foundational to explore the relationships in right-angled triangles and the unit circle, which is a circle with a radius of one unit centered at the origin of a coordinate system. Each point on the unit circle corresponds to an angle, with its x-coordinate representing the cosine, and the y-coordinate representing the sine of that angle.

For a given angle \(\theta\), the basic trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions help to relate the angles of a triangle to the lengths of its sides. Particularly on the unit circle, the functions give specific values:

• \(\sin(\theta)\) corresponds to the y-coordinate of that angle's point on the circle.
• \(\cos(\theta)\) corresponds to the x-coordinate.
• \(\tan(\theta)\) is the ratio \(\frac{\sin(\theta)}{\cos(\theta)}\).

Moreover, understanding these functions allow for the exploration of wave and oscillatory movements, since these functions can model cycles found in nature and engineering.

Angle Reduction

Angle reduction refers to the process of converting larger angles into equivalent angles that fall within the range of 0-360 degrees, or 0-2\(\pi\) radians. This method is based on the fact that the trigonometric functions are periodic, meaning that they repeat their values in regular cycles. Since a full rotation is 360 degrees or 2\(\pi\) radians, any angle that is a multiple of this rotation will lead to the same position on the unit circle.

When you have an angle like 540 degrees, it's an indication that the terminal side of the angle has completed one full revolution (360 degrees) and continued an additional 180 degrees. By reducing the angle, you subtract 360 degrees to find the equivalent position on the unit circle, which simplifies calculation and visualization of the problem. In the given exercise, reducing the angle 540 degrees results in an equivalent angle of 180 degrees.

Reciprocal Trigonometric Identities

Reciprocal trigonometric identities are equations that express the relationship between the basic trigonometric functions and their respective reciprocals. For sine, cosine, and tangent, their reciprocals are cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)), respectively. These identities are immensely useful for simplifying complex trigonometric expressions, solving equations, and proving trigonometric theorems.

The reciprocal identities are given by:

• \(\csc(\theta) = \frac{1}{\sin(\theta)}\), unless \(\sin(\theta)\) is zero.
• \(\sec(\theta) = \frac{1}{\cos(\theta)}\), unless \(\cos(\theta)\) is zero.
• \(\cot(\theta) = \frac{1}{\tan(\theta)}\), unless \(\tan(\theta)\) is zero.

As seen in the example with 180 degrees, the sine value is zero, making the cosecant undefined because division by zero is not possible. Similarly, if the tangent is zero, then the cotangent is also undefined. Mastering these identities is critical in the study of trigonometry and often simplifies finding solutions to various problems.