Exercise 1

Express the following angles in degrees:

1 3 rad

2 \frac { 2 \pi }{ 5 } rad.

3 \frac { 3 \pi }{ 10 } rad.

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Exercise 2

Express the following angles in radians:

1316°

2 10°

3 127º

Exercise 3

Calculate the trigonometric ratios of the following angles:

1 225°

2 330°

3 2,655°

4  -840º

5 -150º

6 1,740°

 

Solution of exercise 1

Express the following angles in degrees:

1 3 rad

\frac { \pi }{ 3 } = \frac { 180° }{ \alpha } \qquad \alpha = \frac { 180 . 3 }{ \pi } = 171.887° = 171° 53'14''

0.887° . 60 = 53.24' \qquad 0.24' \times 16 = 14''

 

2 \frac { 2 \pi }{ 5 } rad.

\frac { 2 \pi }{ 5 } = \frac { 2 \times 180° }{ 5 } = 72°

 

3 \frac { 3 \pi }{ 10 } rad.

\frac { 3 \pi }{ 10 } rad = \frac { 3 \times 180° }{ 10 } = 54°

 

Solution of exercise 2

Express the following angles in radians:

1316°

\frac { \pi }{ \alpha } = \frac { 180° }{ 316° } \qquad \alpha = \frac { 316 \pi }{ 180 } = \frac { 79 \pi }{ 45 } rad

 

2 10°

\frac { \pi }{ \alpha } = \frac { 180° }{ 10° }  \qquad \alpha = \frac { 10 \pi }{ 180 } = \frac { \pi }{ 18 } rad

 

3 127º

\frac { \pi }{ \alpha } = \frac { 180 }{ 127 } \qquad \alpha = \frac { 127 \pi }{ 180 } = 2.216 rad

 

Solution of exercise 3

Calculate the trigonometric ratios of the following angles without using a calculator: (all the angles are in degree)

1 225°

 

\sin { (225°) } = \sin { (45°) } = -\frac { \sqrt { 2 } }{ 2 }

\cos { (225°) } = \cos { (45°) } = -\frac { \sqrt { 2 } }{ 2 }

\tan { (225°) } = \tan { (45°) } = 1

Since the angle is in third quadrant, only \tan { (225) } will be positive.

 

2 330°

 

\sin { (330°) } = \sin { (30°) } = -\frac { 1 }{ 2 }

\cos { (330°) } = \cos { (30°) } = \frac { \sqrt { 3 } }{ 2 }

\tan { (330°) } = \tan { (30°) } = -\frac { \sqrt { 3 } }{ 3 }

Since the angle is in fourth quadrant, only \cos { (330) } will be positive.

 

3 2,655°

 

\sin { (2,655°) } = \sin { (45°) } = \frac { \sqrt { 2 } }{ 2 }

\cos { (2,655°) } = \cos { (45°) } = -\frac { \sqrt { 2 } }{ 2 }

\tan { (2,655°) } = \tan { (45°) } = -1

Since the angle is in second quadrant, only \sin { (2,655°) } will be positive.

 

4  -840º

 

\sin { (-840°) } = \sin { (60°) } = -\frac { \sqrt { 3 } }{ 2 }

\cos { (-840°) } = \cos { (60°) } = -\frac { 1 }{ 2 }

\tan { (-840°) } = \tan { (60°) } =\sqrt { 3 }

Since the angle is in third quadrant, only \tan { (-840°) } will be positive.

 

5 -150º

 

\sin { (-150°) } = \sin { (30°) } = -\frac { 1 }{ 2 }

\cos { (-150°) } = \cos { (30°) } = -\frac { \sqrt { 3 } }{ 2 }

\tan { (-150°) } = \tan { (30°) } = \frac { \sqrt { 3 } }{ 3 }

Since the angle is in third quadrant, only \tan { (-150°) } will be positive.

 

6 1,740°

 

\sin { (1,740°) } = \sin { (60°) } = -\frac { \sqrt { 3 } }{ 2 }

\cos { (1,740°) } = \cos { (60°)} = \frac { 1 }{ 2 }

\tan { (1,740°) } = \tan { (60°) } = - \sqrt { 3 }

Since the angle is in fourth quadrant, only \cos { (1,740°) } will be positive.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.