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Trigonometry

Signs of Trigonometric Functions

As discussed in the lesson for Angles in Standard Position, there are four quadrants on the Cartesian plane: 

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Recall our three trigonometric functions and their reciprocals: 

                                       \( f(\theta) = sin \theta = \frac{y}{r} \)                                           \( F(\theta) = csc \theta = \frac{1}{sin \theta} = \frac{r}{y} \)

                                       \( g(\theta) = cos \theta = \frac{x}{r} \)                                           \( G(\theta) = sec \theta = \frac{1}{cos \theta} = \frac{r}{x} \)

                                       \( h(\theta) = tan \theta = \frac{y}{x} \)                                          \( H(\theta) = cot \theta = \frac{1}{tan \theta} = \frac{x}{y} \)

 

Depending on the angle (between 0° and 360°, or 0 and 2\( \pi \)), these functions will return a positive or a negative value. The following diagram shows which sign the output will be as it pertains to the quadrant where the terminal arm lies.

Tip: Use the CAST rule to remember the quadrants where each function is positive. undefined

 

                                         Cosine and secant 

                                All 

                                Sine and cosecant 

                                Tangent and cotangent 

 

 

Examples 

a) Kira is given an angle of 220° from the initial side that is the positive x-axis. What is the tangent ratio corresponding to this angle, and is it positive or negative?

b) Franklin is told to draw an angle that produces a negative cosine ratio and a negative tangent ratio. What are the least and greatest angles that Franklin can draw? Express your answer in degrees and radians. 

See the video below for the solutions:

The Unit Circle

The unit circle is a circle with radius of 1 unit centered at the origin of the Cartesian plane. 

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In trigonometry, the x-coordinates are represented by the values of \(cos\theta \) and the y-coordinates are represented by the values of \(sin\theta \). \( \theta \) refers to the angle that lies in between the positive x-axis (the initial side) and the terminal arm going through a particular point on the unit circle. 

There are three particular angles which make up the unit circle, commonly referred to as the special angles of the special triangles that contain them:

 

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If you look closely at the angles in radians on the unit circle, you may notice that they all stem from one of the three special angles. Given the special triangles and the unit circle, finding values of \( sin\theta \) and \( cos\theta \) is fairly straightforward. 

What if you are given the values, and not the angles? Luckily, we have the reference angle approach that can help us to determine the angles we are missing. 

See the video below for the explanation and examples:

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