Question

Find the exact values of the trigonometric functions for \(\theta=120^{\circ} .\)

Short answer

The exact values of the trigonometric functions for \(\theta = 120^{\circ}\) are: \[\sin \theta = \frac{\sqrt{3}}{2}, \quad \cos \theta = -\frac{1}{2}, \quad \tan \theta = -\sqrt{3}\] \[\csc \theta = \frac{2\sqrt{3}}{3}, \quad \sec \theta = -2, \quad \cot \theta = -\frac{\sqrt{3}}{3}\]

Step-by-step solution

Determine the reference angle

Since \(\theta = 120^{\circ}\) is in the second quadrant, we can find a reference angle \(\alpha = 180^{\circ}-120^{\circ}=60^{\circ}\). -Step 2: Determine the coordinates on the reference triangle

Determine the coordinates on the reference triangle

The reference angle \(\alpha\) can be used to find the sides of a 30-60-90 reference triangle. In a 30-60-90 triangle, the sides are proportional to the ratios \(1 : \sqrt{3} : 2\). Since \(\alpha = 60^{\circ}\), we have: Adjacent side length = 1 (smallest side, since it's opposite the smallest angle) Opposite side length = \(\sqrt{3}\) (since it is opposite the angle \(\alpha\)) Hypotenuse = 2 -Step 3: Compute sine, cosine, and tangent

Compute sine, cosine, and tangent

Using the coordinates and side lengths, we can compute the three main trigonometric functions: \[\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}\] \[\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = -\frac{1}{2}\] (Negative, since we're in the second quadrant) \[\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{-1} = -\sqrt{3}\] -Step 4: Compute the reciprocal functions

Compute the reciprocal functions

Using the values from the previous step, we can calculate the reciprocal functions: \[\csc \theta = \frac{1}{\sin \theta} = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\] \[\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{1}{2}} = -2\] \[\cot \theta = \frac{1}{\tan \theta} = -\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = -\frac{\sqrt{3}}{3}\] Thus, the exact values of the trigonometric functions for \(\theta = 120^{\circ}\) are: \[\sin \theta = \frac{\sqrt{3}}{2}, \quad \cos \theta = -\frac{1}{2}, \quad \tan \theta = -\sqrt{3}\] \[\csc \theta = \frac{2\sqrt{3}}{3}, \quad \sec \theta = -2, \quad \cot \theta = -\frac{\sqrt{3}}{3}\]

Key concepts

Reference Angle

The reference angle is a helpful concept in trigonometry. It is the acute angle formed between the terminal side of an angle and the horizontal axis. When dealing with angles larger than 90 degrees, finding the reference angle allows us to easily calculate trigonometric functions by relating them to known triangle ratios.
For an angle of \(\theta = 120^\circ\), it lies in the second quadrant. To find its reference angle, we subtract it from \(180^\circ\): \(\alpha = 180^\circ - 120^\circ = 60^\circ\).
This reference angle of \(60^\circ\) is important because it allows us to use familiar values from specific triangles to find exact trigonometric values.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions provide functions that are the inverse of the standard sine, cosine, and tangent functions. Let's take a closer look at what each reciprocal function is:

• **Cosecant (\(\csc\)):** The reciprocal of sine. \[\csc \theta = \frac{1}{\sin \theta}\]
• **Secant (\(\sec\)):** The reciprocal of cosine. \[\sec \theta = \frac{1}{\cos \theta}\]
• **Cotangent (\(\cot\)):** The reciprocal of tangent. \[\cot \theta = \frac{1}{\tan \theta}\]

Computing these recurrences help understand deeper relationships in trigonometry and is useful in many trigonometric equations and solving tasks. Especially, in this example scenario where our angle is \(120^\circ\), lying in the second quadrant, these reciprocal values can be quickly calculated using the acquired values of sine, cosine, and tangent.

30-60-90 Triangle

The 30-60-90 triangle is a special type of right triangle that has angles measuring \(30^\circ, 60^\circ,\) and \(90^\circ\). This triangle is unique because the side lengths are in a fixed ratio: \(1 : \sqrt{3} : 2\). Knowing this ratio is beneficial because it allows us to easily find the lengths of sides when the value of one side is known.
The shortest side, opposite the \(30^\circ\) angle, has a length of 1. The side opposite the \(60^\circ\) angle has a length of \(\sqrt{3}\). The hypotenuse, opposite the \(90^\circ\) angle, has a length of 2.
In our problem where \(\theta = 120^\circ\), the reference angle is \(60^\circ\). Therefore, the triangle's properties are optimally applicable, and we can easily determine the values of sine, cosine, and tangent.

Second Quadrant

Understanding which quadrant an angle lies in helps determine the signs of its trigonometric functions. The coordinate plane is divided into four quadrants:

• **First Quadrant:** All trigonometric values are positive.
• **Second Quadrant:** Sine is positive, while cosine and tangent are negative.
• **Third Quadrant:** Tangent is positive, while sine and cosine are negative.
• **Fourth Quadrant:** Cosine is positive, while sine and tangent are negative.

When \(\theta = 120^\circ\), it clearly places the angle in the second quadrant. This tells us that the sine value is positive, but cosine and tangent values are negative. Observing such pattern assist with quick checks on the calculated results' consistency.

Exact Trigonometric Values

Exact trigonometric values refer to the precise values that trigonometric functions have at specific angles, especially those most commonly encountered, like \(0^\circ, 30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\). Calculating these values precisely instead of approximately is crucial for many mathematical problems.
In the given exercise, the reference angle \(60^\circ\) allows us to directly apply these exact values:

• **Sine:** \[\sin 60^\circ = \frac{\sqrt{3}}{2}\]
• **Cosine:** \[\cos 60^\circ = \frac{1}{2}\] (but negative in the second quadrant, so \(\cos 120^\circ = -\frac{1}{2}\))
• **Tangent:** \[\tan 60^\circ = \sqrt{3}\] (again, negative in the second quadrant, \(\tan 120^\circ = -\sqrt{3}\))

Knowing these exact values allows us to define reciprocal functions precisely and aids in solving various real-world problems where estimation simply isn't sufficient.