Trigonometric identities of other angles

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: $$240^\circ-60^\circ=180^\circ$$

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:

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

2. We then take the residual. In the previous example it is $60^\circ$.

3. 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 $320^\circ-50^\circ=270^\circ$.

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:

• $\sin(320^\circ)=-\cos(50^\circ)$

• $\cos(320^\circ)=\sin(50^\circ)$

• $\tan(320^\circ)=-\cot(50^\circ)$