CAL-PREP TRIG REVIEW-2

DEFINITION:

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

it takes
radians to make a full circle.

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

 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°
° 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 y coordinates of the points on this circle are the cosine and sine of the angle formed between the radius of the circle and the positive x-axis,

Note: sin u > 0 in the 1st & 2nd quadrants since
sin u is the y-value at any point on the circle and y > 0 in those quadrants.

cos u > 0 in the 1st & 4th quadrants since cos u is the x-value at any point on the circle
and x > 0 in the 1st and 4th quadrants.

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,
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)

.

2)

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

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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