Negative Angles Used In Trigonometric Functions

In trigonometry, we are dealing with angles. The value of the angles can be indicated as positive or negative. What does that mean?

Convention:

Positive-valued angle means anti-clockwise rotation (or forward moving direction).

Negative-valued angle means clockwise rotation (or reverse moving direction).

Diagram 1 below shows the convention in graphic form.

Diagram 1

Besides rotational direction, what does the "sign" of the angle has on the numerical value of their trigonometric operation?

Is cos 60⁰ = cos (- 60⁰) ?

How about its sine counterpart?
Is sin 60⁰ = sin (- 60⁰) ?

The diagrams below will provide the answers to the above two questions.


Diagram 2: Cosine function

In diagram 2, we can see that the amplitude of the cosine for +60⁰ and -60⁰ remains the SAME.

That is to say, whether the angles are positive or negative, cosine function of these angles are the same.

Let's look at the Sine operation of negative angles (Diagram 3).



Diagram 3: Sine function

Here in diagram 3, we see a difference. The numerical value of sine operation on positive and negative angle differs by the sign. They are DIFFERENT!

sin 60⁰ = - sin (-60⁰)

In conclusion,
cos A = cos (-A) and
sin A = - sin (-A).
As such tan A = - tan (-A) since tan A = sin A / cos A.

"-" and "+" can be confusing.
However, do note that it is the learner's duty to master this concept.

Pressurised? Fear not!
Mindset change will do the trick. "I love maths as it is interesting!"

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