# Problems from Trigonometric identities of other angles

Determine whether the following equations/identities are true:

• $$\sin(75^\circ)=\sin(105^\circ)$$

• $$\tan(220^\circ)=-\tan(40^\circ)$$

• $$\cos(350^\circ)=-\cos(170^\circ)$$
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### 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.
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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.

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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.

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