Trigonometric identities of other angles
Determine whether the following equations/identities are true:
$\sin(75^\circ)=\sin(105^\circ)$
$\tan(220^\circ)=-\tan(40^\circ)$
$\cos(350^\circ)=-\cos(170^\circ)$
An angle of $75^\circ$ and one of $105 ^\circ$ are supplementary, since $75^\circ+105 ^\circ=180^\circ$. Since the sines of supplementary angles are equal, the identity is true.
An angle of $220^\circ$ and one of $40^\circ$ differ in $180^\circ$, because $220^\circ-40^\circ=180^\circ$. Since the angles that differ in $180^\circ$ have the same tangent, then the equation is false.
An angle of $350^\circ$ and one of $170^\circ$ differ in $180^\circ$, since $350^\circ-170^\circ =180^\circ $. The cosines of angles that differ in $180^\circ$ have equal cosines, but with a different sign. That is: $\cos(350^\circ)=-\cos(170^\circ)$, therefore the identity is true.
- The identity is true.
- The identity is false.
- The identity is true.
Are the following identities or statements correct?
a) $\sin(45^\circ)=-\sin(315^\circ)$
b) The angles $80^\circ$ and $100^\circ$ are opposites.
c) $\tan(-17^\circ)=\tan(17^\circ)$
d) $\cos(450^\circ)=\cos(90^\circ)$
a) The angles $45^\circ$ and $315^\circ$ are opposites, since $45^\circ+315^\circ=360^\circ$. Since the sine of two opposite angles is equal but with a different sign, it is has to be the case that, $\sin(45^\circ)= -\sin(315^\circ)$. Thus, the identity is correct.
b) The angles $80^\circ$ and $100^\circ$ add up to $ 80^\circ+ 100^\circ =180^\circ$. Therefore they are not opposites, since they do not add up to $360^\circ$. In fact, they are supplementary.
c) he tangent of a negative angle is the same as that of the positive angle, but with the opposite sign. In this case it means that: $\tan(-17^\circ)=-\tan(17^\circ)$. Therefore, the identity is false.
d) If we subtract $450^\circ$ from $360^\circ$, we have $90^\circ$ left. Therefore, the cosines of these two angles are the same: $\cos(450^\circ)=\cos(90^\circ)$. The identity is, then, correct.
a) The identity is correct.
b) The statement is false.
c) The identity is false.
d) The identity is correct.
Are the following identities correct?
a) $\tan(37^\circ)=-\cot(233^\circ)$
b) $\cos(400^\circ)=-\cos(130^\circ)$
c) $\cos(230^\circ)=-\sin(140^\circ)$
a) $37^\circ$ and $233^\circ$ add up to $270^\circ$: $$37^\circ+233^\circ=270^\circ$$
Thus, we have: $\tan(37^\circ)=\cot(233^\circ)$ (and not with a minus sign). Therefore the identity is false.
b) The angles $400^\circ$ and $130^\circ$ differ in $270^\circ$: $$400^\circ-130^\circ=270^\circ$$
Thus, we have: $\cos(400^\circ)=\sin(130^\circ)$, therefore the identity is false.
c) The angles $230^\circ$ and $140^\circ$ differ in $90^\circ$, since $$230^\circ-140^\circ=90^\circ$$
Thus, we have: $\cos(230^\circ)=-\sin(140^\circ)$. Therefore the identity is correct.
a) The identity is false.
b) The identity is false.
c) The identity is correct.