Supplementary angles
Two angles are said to be supplementary if they add up to $$180^\circ$$.
For example, an angle of $$140^\circ$$ and one of $$40^\circ$$ are supplementary since: $$$140^\circ+ 40^\circ=180^\circ $$$
The sine, cosine and tangent of the supplementary angles have a certain relation. That is, if $$\alpha$$ and $$\beta$$ are two supplementary angles then we have:

$$\sin(\alpha)=\sin(\beta)$$

$$\cos(\alpha)=\cos(\beta)$$
 $$\tan(\alpha)=\tan(\beta)$$
So we have that their sines are equal, and their cosine and their tangent are equal with opposed signs.
In the previous example, then, we have:

$$\sin(40^\circ)=\sin(140^\circ)$$

$$\cos(40^\circ)=\cos(140^\circ)$$
 $$\tan(40^\circ)=\tan(140^\circ)$$
Angles that differ in $$180^\circ$$
Two angles $$\alpha$$ and $$\beta$$ are said to differ in $$180^\circ$$ if $$\alpha\beta=180^\circ$$.
For example an angle of $$240^\circ$$ and one of $$60^\circ$$ differ in $$180^\circ$$, since: $$$240^\circ60^\circ=180^\circ$$$
The sine, cosine and tangent of two angles that differ in $$180^\circ$$ are also related. If $$\alpha$$ and $$\beta$$ differ in $$180^\circ$$, we have:

$$\sin(\alpha)=\sin(\beta)$$

$$\cos(\alpha)=\cos(\beta)$$
 $$\tan(\alpha)=\tan(\beta)$$
That is, the sine and the cosine have equal values but differ in their signs, while the tangent is equal.
In the previous example, therefore, we have:

$$\sin(240^\circ)=\sin(60^\circ)$$

$$\cos(240^\circ)=\cos(60^\circ)$$
 $$\tan(240^\circ)=\tan(60^\circ)$$
Opposite angles
Two angles are said to be opposite angles if they add up to $$360^\circ$$. That is, $$\alpha$$ and $$\beta$$ are opposite angles if $$\alpha+\beta=360^\circ$$.
For example, an angle of $$330^\circ$$ and one of $$30^\circ$$ are opposite angles, since $$$330^\circ+30^\circ=360^\circ$$$
The sines, cosines and tangent of opposite angles are related in a similar way as the one we saw with the supplementary angles or those which differ in $$180^\circ$$. That is, if $$\alpha$$ and $$\beta$$ are opposite angles we have:

$$\sin(\alpha)=\sin(\beta)$$

$$\cos(\alpha)=\cos(\beta)$$
 $$\tan(\alpha)=\tan(\beta)$$
That is, the sine and the tangent are equal but with different signs, and the cosine is exactly the same.
In the previous example we have:

$$\sin(330^\circ)=\sin(30^\circ)$$

$$\cos(330^\circ)=\cos(30^\circ)$$
 $$\tan(330^\circ)=\tan(30^\circ)$$
Negative angles
An angle is negative if it goes clockwise, and it is symbolized by a minus sign.
For example, if there is an angle of $$30^\circ$$, but instead of going up it goes down, or clockwise, it is said that the angle is of $$30^\circ$$.
The following illustration shows the negative angle $$30^\circ$$:
If $$\alpha$$ is an angle, then we have the following identities:

$$\sin(\alpha)=\sin(\alpha)$$

$$\cos(\alpha)=\cos(\alpha)$$
 $$\tan(\alpha)=\tan(\alpha)$$
In short, the sine and the tangent of $$\alpha$$ and $$\alpha$$ are the same but with different signs, and the cosine is exactly the same.
In the previous example we have:

$$\sin(30^\circ)=\sin(30^\circ)$$

$$\cos(30^\circ)=\cos(30^\circ)$$
 $$\tan(30^\circ)=\tan(30^\circ)$$
Angles greater than $$360^\circ$$
To find the sine, the cosine and the tangent of an angle greater than $$360^\circ$$, we have to do the following:

The integer division of the given angle over $$360$$. For example, if the angle is $$780^\circ$$, then:

We then take the residual. In the previous example it is $$60^\circ$$.
 The sine, the cosine and the tangent of the given angle are that of the residual that has been obtained.
Going back to the previous example, we have:

$$\sin(780^\circ)=\sin(660^\circ)$$

$$\cos(780^\circ)=\cos(60^\circ)$$
 $$\tan(780^\circ)=\tan(60^\circ)$$
Angles that differ in $$90^\circ$$
Two angles differ in $$90^\circ$$ if the result of subtracting them is $$90^\circ$$.
For example, an angle of $$160^\circ$$ and one of $$70^\circ$$, ja que: $$160^\circ 70^\circ= 90^\circ$$. The following illustration shows it more clearly:
If it is true that two angles, $$\alpha$$ and $$\beta$$, differ in $$90^\circ$$ (that is to say, if $$\alpha\beta=90^\circ$$) then:

$$\sin(\alpha)=\cos(\beta)$$

$$\cos(\alpha)=\sin(\beta)$$
 $$\tan(\alpha)=\cot(\beta)$$
In the previous example we have that:

$$\sin(160^\circ)=\cos(70^\circ)$$

$$\cos(160^\circ)=\sin(70^\circ)$$
 $$\tan(160^\circ)=\cot(70^\circ)$$
Angles that add up to $$270^\circ$$
Two angles $$\alpha$$ and $$\beta$$ add up $$270^\circ$$ if $$\alpha+\beta=270^\circ$$.
For example, an angle of $$70^\circ$$ and one of $$200^\circ$$, since $$70^\circ + 200^\circ=270^\circ$$.
In this case, $$\alpha$$ and $$\beta$$ satisfy the following identities:

$$\sin(\alpha)=\cos(\beta)$$

$$\cos(\alpha)=\sin(\beta)$$
 $$\tan(\alpha)=\cot(\beta)$$
In the previous example, we have:

$$\sin(70^\circ)=\cos(200^\circ)$$

$$\cos(70^\circ)=\sin(200^\circ)$$
 $$\tan(70^\circ)=\cot(200^\circ)$$
Angles that differ in $$270^\circ$$
Two angles $$\alpha$$ and $$\beta$$ differ in $$270^\circ$$ if, when subtracted, we obtain $$270^\circ$$: $$\alpha\beta= 270^\circ$$.
An example is the angles of $$320^\circ$$ and $$50^\circ$$, since $$320^\circ50^\circ=270^\circ$$.
When two angles $$\alpha$$ and $$\beta$$ differ in $$270^\circ$$ we have:

$$\sin(\alpha)=\cos(\beta)$$

$$\cos(\alpha)=\sin(\beta)$$
 $$\tan(\alpha)=\cot(\beta)$$
In our example with the angle of $$320^\circ$$ and $$50^\circ$$, we have:

$$\sin(320^\circ)=\cos(50^\circ)$$

$$\cos(320^\circ)=\sin(50^\circ)$$
 $$\tan(320^\circ)=\cot(50^\circ)$$