Problems from Trigonometric identities of other angles

Determine whether the following equations/identities are true:

$\sin (75^{\circ}) = \sin (105^{\circ})$
$\tan (220^{\circ}) = - \tan (40^{\circ})$
$\cos (350^{\circ}) = - \cos (170^{\circ})$

See development and solution

Development:

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.

Solution:

The identity is true.
The identity is false.
The identity is true.

Hide solution and development

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) $\cos (450^{\circ}) = \cos (90^{\circ})$

See development and solution

Development:

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.

Solution:

a) The identity is correct.

b) The statement is false.

c) The identity is false.

d) The identity is correct.

Hide solution and development

Are the following identities correct?

a) $\tan (37^{\circ}) = - \cot (233^{\circ})$

b) $\cos (400^{\circ}) = - \cos (130^{\circ})$

c) $\cos (230^{\circ}) = - \sin (140^{\circ})$

See development and solution

Development:

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.

Solution:

a) The identity is false.

b) The identity is false.

c) The identity is correct.

Hide solution and development

View theory