CAL-PREP TRIG REVIEW-2 |

__RADIANS__:

__DEFINITION__:

A **radian** **is an angle** at the center of a circle which **subtends an ****arc equal to the radius**.

Since **each radian defines an arc = the radius**, and, ,

it takes **radians to make a full circle**.

Therefore, there are ** radians in half a circle**, a **straight line** or 180°.

**RADIAN/DEGREE EQUIVALENCES:**

^{R} = 360° |
^{R} = 180° |
^{R} = 90° |
^{R} = 45° |
^{R} = 60 ° |

^{ R} = 30° |
^{R} = 20° |
^{R} = 15° |
1^{ R} = ° |
1° = ^{ R} |

all values come from the basic relationship ^{R} = 180°

and **1 radian = 57.29578 **° or 57.3°

**CHANGING FROM ONE SYSTEM TO THE OTHER:**

To find the radian equivalence of 120°, we use 1° = ^{ R}.

Multiplying both sides by 120, we get: 120°=

A better way uses the fact that 120° = 2 times 60°, and we know 60° = ^{R},

so we multiply by 2 to get 120°, which gives us .

The same holds true for any multiple of the basic values.

For the radian measure of 135°: since 135° = 3 times 45° and 45° = ^{R}, 135° = ^{R} .

The same is true when **changing from radians to degrees**.

Since ^{R} = 180°, we can always replace with 180.

So, if we need the degree equivalence of radians, we **replace **** with 180**,

to get **180 / 15 = 12°**.

If there is no in the radian measure, we use 1 radian = degrees.

**Example:** Find the degree measure of 3.5 radians.

Multiply both sides by 3.5 to get: 3.5^{ R} =

** Note**: when changing from degrees to radians, leave in the expression.

When going the other way, (from radians to degrees), enter in your calculator (as I did in the previous example), to get a number value.

**ARC LENGTH:**

Since **the measure of an angle in radians determines the length of the arc **subtended by the angle, we can find the length of the arc created by an angle if we know the radius of rotation.

** **

**Example:** Find the length of the arc subtended by an angle of 2.4 radians at the center of a circle with a radius of 4 cm.

Since , then **L = 2.4 (4) = 9.6 cm.**

If the **angle** is given **in degrees**, **change** it **to radians** then find the arc length.

If given the arc length **L** and the radius ** r**, we find in radians.

By the way, this is how the odometer works in a car or on a bicycle.

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**THE UNIT CIRCLE :**

The unit circle is a circle, center (0, 0) with a **radius of 1** unit.

The** **** x** and

**Note:** sin *u* > 0 in the 1st & 2nd quadrants since

**sin u is the y-value** at any point on the circle and

cos *u* > 0 in the 1st & 4th quadrants since **cos u is the x-value**

and

tan *u* > 0 when *x* and *y* have the same sign, ie: in the 1st and 3rd quadrants.

cos 0 = 1 | sin 0 = 0 | tan 0 = 0 |

cos = 0 | sin = 1 | tan (asymptote) |

cos = – 1 | sin = 0 | tan o = 0 |

cos = 0 | sin = – 1 | tan (asymptote) |

the values for are the same as those for 0.

We can now write the coordinates of a point (*x*, *y*) as (cos *A* , sin *A* ).

The cosine and sine functions have domain ** R**, and range [ – 1, 1 ].

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__Points on the Unit Circle__:

We use the triangles to find the coordinates of the 12 points on the circle.

The cosine and sine values of 30°, 60°, and 45° are read from the triangles.

P() = ( 0, 1 ) |

From the symmetry of the unit circle, and our knowledge of the signs for the *x* and *y* values, we find the remaining coordinates.

** Note**: we generally express the sine and the cosine of 45° as rather than .

**Hints and Notes:**

**Note1: **A full rotation = radians = = = is a full circle.

So an angle of is in the 1st quadrant since it's radians greater than a full circle.

**Note2:** To locate an angle of , just count your way to the proper quadrant.

There are 3 angles = per quadrant so takes us to the 2nd quad. leaving

one more angle = .

**Note3:** We read the sign (positive or negative) of the trig function from the quadrant.

**Example 1:**

The coordinates of P(A), a point on the unit circle are .

Find 2 **coterminal angles** for A.

We've got a multiple of 45° or radians. Since *y* < 0, but *x > 0* it's in the 4th quad.

It takes 7 angles of 45° or radians to reach the 4th quad,

so the angle is either or . It could also be or .

**Example 2:**

Find the 6 trig functions values for angle A =

Since we have a multiple of , we use the 30°, 60°, 90° triangle,

then adjust the sign to fit the quadrant.

is * more than* a complete rotation clockwise.

So it's **in the 3rd quadrant** where only tan A is positive.

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**Practice**

1) For the angles listed below:

a) indicate the **quadrant** in which the angle is found

b) convert to **radian measure**. Do not use 3.14 for

^{}

i) 120° | ii) 405° | iii) 225° | iv) 375° | v) -150° |

2) For the angles listed below:

a) indicate the quadrant in which the angle is found

b) convert to degree measure.

i) | ii) | iii) | iv) | v) |

3) The wheels of a bicycle have a 24-inch **diameter**. If they make 12 rotations per minute, how far will the bike travel in 3 minutes? Give the answer in **feet** to the nearest tenth. (12 in. = 1 foot)

4) Complete the following table without a calculator: do not rationalize!!*
*(you can print out the table to do the question -- just the table!!

**Hint**: a / 0 = -- divide by zero, get infinity.

A | sin A | cos A | tan A | csc A | sec A | cot A |

0 |

5) State 2 coterminal angles for:

a) P(A) = (0, – 1) | b) |

6) Evaluate:

a) | b) |

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**Solutions**

1)

i) 120° 2nd quadrant |
ii) 405° 1st quadrant |
iii) 225° 3rd quadrant |
iv) 375° 1st quadrant |
v) – 150° 3rd quadrant |

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2)

i) 2nd quadrant 135° |
ii) 1st quadrant 72° |
iii) 4th quadrant 1020° |
iv) on negative
-2700° |
v) 3rd quadrant – 150° |

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3) Since the diameter = 2 ft., the radius = 1 ft. (24 inches = 2 feet for metric users)

Since 1 rotation/min = radians/min, 12 rotations/min = radians/min.

Therefore, in **3 minutes** the wheel travels through ** radians**.

So, **the bike will travel **** feet = 226.2 feet**.

4)

A | sin A | cos A | tan A | csc A | sec A | cot A |

– 1/2 | – 2 | |||||

–1 | 0 | – 1 | 0 | |||

–1/2 | –2 | |||||

–1 | –1 | |||||

0 | –1 | 0 | –1 | |||

0 | 0 | –1 | 0 | 1 |

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5) a)

b)

6)

a) = 2(1) + 3( – ½) – 5( – 1) = 5.5 |

b) |

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