TRIGONOMETRY
 

Objectives: To learn what sine and cosine mean.

Recall 1: Sin(a) is the height of a right triangle that has a hypotenuse of length 1 unit and base angle a radians.
Cos(a) is the base of a right triangle that has a hypotenuse of length 1 unit and base angle a radians.

Recall 2:  Imagine that a point is moving in the counterclockwise direction around a circle of radius 1.

Can you imagine the height of the point?  Can you imagine the y-coordinate of the point?
Can you imagine the x-coordinate of the point?

Recall 3  Suppose that a line joining P with the center of the circle makes an angle a with the diameter.  Then the coordinates of the point P are represented by (cos a, sin a).

Recall 4 The illustration below shows how the sine and cosine functions relate to the coordinates of the point P.   Remember that the circle is of radius 1.

Recall 5 The Pythagorean Theorem tells us that there is a relationship between
sin a and cos a.
 

Recall 6: From this we can find the sine and cosine of typical angles like pi/6,  pi/4, pi/3 and pi/2.  Clearly:

sin pi/2 = 1 and cos pi/2 = 0
sin 0 = 0 and cos 0 = 1

To find the sine and cosine of angles like pi/6,  pi/4 and pi/3 draw an equilateral triangle with sides of length 1.

Next, divide the triangle in half by dropping a perpendicular line from the top.  (See illustration below.)

From this you should be able to see that cos(pi /3) = 1/2 and that sin(pi/6) = 1/2.
Do you see why?  Now find the height of the triangle by using the Pythagorean Theorem on the left hand side right triangle.  The height is sqrt(3)/2.   Do you see why?

Recall 7: You now know that sin(pi/3) = sqrt(3)/2 and cos(pi/6) = sqrt(3)/2.

Recall 8:    You could find the value of sin(pi/4) by taking a square with sides 1 unit and cutting it by the diagonal.  Then sin(pi/4) = sqrt(2)/2 and cos(pi/4) = sqrt(2)/2.

The sine and cosine functions enable us to find one side of a right triangle simply from information about one angle and one other side.   For example, if you are asked to find the height of a triangle that has a base angle pi/3 radians and base 4 units, you would draw the triangle and notice that the height is just 4sin(pi/3).  And since sin( pi/3) = sqrt(3)/2, you would know that the height is 2sqrt(3).

 You will find that many problems in nature involve situations where an angle and a length is known but another length is needed.
For example, if you wanted to find the height of a tower, just move, say, 1000 meters from the tower and measure the angle of your line-of-site to the top of the tower.  The height is just 1000 times the sine of the angle of your line-of-site.

1. Whenever you have a problem that involves right triangles, the Pythagorean Theorem may be of help.  Recall that it says that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse.  In the triangle below, a^2 + b^2 = c^2.

 

2.  The Pythagorean Theorem also gives a relationship between the sine and cosine functions.

(sin a)^2 + (cos a)^2 = 1.

To see this draw a right triangle with hypotenuse 1 and base angle a.  Then the height is sin a and the base is cos a.

3. An obvious property of sines and cosines is that their values lie between -1 and 1. Every point on the unit circle is 1 unit from the origin, so the coordinates of any point are within 1 of 0 as well.

 |sin a| <= 1 and |cos a| <= 1.

4.
 sin 0 = 0          cos 0 = 1
 sin pi/6 = 1/2     cos pi/6 = *3/2
 sin pi/4 = *2/2   cos pi/4 = *2/2
 sin pi/3 = *3/2   cos pi/3 = 1/2
 sin pi/2 = 1       cos pi/2 = 0

5. Notice that sin a is negative for a greater than pi and less than 2pi.  Cos a is negative for a greater than pi/2 and less than 3pi/2.

6.   Sine is an odd function, and cosine is an even function.

You may not have come across these adjectives "odd" and "even" when applied to functions, but it's important to know them.

• A function f is said to be an odd function if for any number x, f(-x) = -f(x).
• A function f is said to be an even function if for any number x, f(-x) = f(x).

Sine is an odd function, and cosine is even:
 

These facts follow from the symmetry of the unit circle across the x-axis. The angle ?a is the same angle as a except it's on the other side of the x-axis. Flipping a point (x,y) to the other side of the x-axis makes it into (x,-y), so the y-coordinate is negated, that is, the sine is negated, but the x-coordinate remains the same, that is, the cosine is unchanged.

7. Sine and cosine are periodic functions of period 360°, that is, of period 2pi. That's because sines and cosines are defined in terms of angles, and you can add multiples of 360°, or 2pi, and it doesn't change the angle. Thus,

sin (a + 360°) = sin a,

cos (a + 360°) = cos a

Many of the modern applications of trigonometry follow from the uses of trig to calculus, especially those applications which deal directly with trigonometric functions. So, we should use radian measure when thinking of trig in terms of trig functions. In radian measure that last pair of equations read as

sin (a + 2pi) = sin a,

cos (a + 2pi) = cos a.
 

8.  Sine and cosine are complementary:

cos a = sin (pi/2 - a)

sin a = cos (pi/2 - a)

We've seen this before, but now we have it for any angle t. It's true because when you reflect the plane across the diagonal line y = x, an angle is exchanged for its complement.

sin (a + pi) = -cos a
cos (a + pi) = -sin a

Example 1.   Suppose you know that cos pi/5 = 0.809017.  Find sin pi/5.

Solution:   Use the Pythagorean Theorem to express the relation between sin pi/5 and
cos pi/5.  Then sin(pi/5)^2 + cos(pi/5)^2 = 1.  This means that

But pi/5 is greater than 0 and less than pi.  Therefore, sin pi/5 must be positive.  Ignore the negative value so sin pi/5 = 0.5877785.

Example 2.   A gable roof above a garage is to be made from 16 feet long rafters.   If the pitch of the roof is to be at least 0.7 radians, find the largest possible width of the garage.

Solution:  Drop a perpendicular line from the top of the garage to split the triangle into two right triangles.   The base of one of the triangles is 1/2 the width of the garage.  So all we need to do is find that base.  Call it x.  Then cos 0.7 = x/16.  Multiply both sides by 16 to get x = 16 cos 0.7 = 12.24 feet.  So the maximum width of the garage is 2x = 24.48 feet.

Example 3.   A bridge makes an angle of pi/3 with a river.  If the width of the river is 30 meters, find the length of the bridge.

Solution: If the river is 30 meters wide, then we may represent the length of the bridge as the hypotenuse of a right triangle whose height is 30.  Let x = the length of the bridge (the hypotenuse).  Then sin (pi/3) = 30/x.  This is the same thing as saying that x = 30/sin (pi/3) = 30/(*3/2) = 34.64 meters

1.  If cos a = 2/3 what is sin a?
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2. A circle of radius 4 is centered at the point (0,0).  An angle centered at the center of a circle cuts the circle at the coordinates  (a,b).  Find a and b.
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3  A right triangle has hypotenuse 5 and base angle pi/3.  Find the measurements of all the sides and angles of the triangle.
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4 A right triangle has sides 4 and 5. Find the measurements of all the sides and angles of the triangle.
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5 A line joining the coordinates (0,0) with (3,-4) makes an angle a with the horizontal line through (0,0).  Find sin a and cos a.
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6 A balloon is at an altitude of 200 feet.  The line of site from an observer is 50°.  How far is the balloon from the observer?
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7 A surveying team measures an angle of pi/8 radians from a point A on a beach to a point B at the top of a cliff.  If point A is 200 meters from the base of the cliff, how high is the cliff?
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8 A crane is 80 feet long.  For safety reasons it should never be lowered less than 55° from the horizontal.   What is the largest distance from its base that the crane can reach?
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9 If you fly 300 miles at 20° measured from an east/west line in going from one place to another and return by car following roads that go only east and south, what is the difference in mileage between traveling by plane and car?
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10 A mine is 2,000 feet in elevation.  The straight 500-foot shaft makes an angle of 20° with the horizontal.  The bottom of the mine is at the lower end of the shaft.  Find the elevation of the bottom of the mine.
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 ANSWERS

1. ±*5/3

2. a = 4 cos a, b = 4 sin a

3. other angle is pi/6 radians
 side opposite pi/3 is 5*3/2
 base is 5/2

4. hypotenuse = *41
 base angle = 0.9 radians
 remaining angle = 0.67 radians

5.  sin a = -4/5
 cos a = 3/5

6. 200/sin 50°feet = 261 feet

7. 82.84 meters

8. 80cos 55°feet = 45.89 feet

9.   84.51 miles

10.  1829 feet