Introduction to Trigonometry
Review for Quiz (old review- not for 2002 but extra practice if you want)
Trigonometry is based on similar right triangles. Since the sides are proportional between similar triangles, the ratios between sides in similar triangles will be the same.
Example:
Notice that the ratio of the side opposite the 30 degree angle to the hypotenuse in both triangles has a ratio of 1/2
Given any Right triangle, an acute angle is chosen which is referred to as a reference angle.
If < A is chosen as the reference angle the side directly across from the angle is referred to as the opposite side.
The side which connects the angle to the right angle of the triangle is referred to as the adjacent side.
The side opposite the right angle is always referred to as the hypotenuse
BC = opposite side AC = the adjacent side AB = hypotenuse
The 3 Basic Trigonometric functions are as follows
To memorize these functions
Remember the king of Trigonometry - SOHCAHTOA
or the phrase : Some old horse came a hoppin through our attic
or if you can remember Sine Cosine and Tangent in order
Once you have these functions memorized, you can solve for missing pieces of a right triangle by setting up the functions
Examples
Since x is the opposite side and 10 is the hypotenuse, you know that you must use the Sine as it is the ratio of the opposite to the hypotenuse. Use the table on page 424 or use your calculator to find the sine of 35 which is .5736. Substitute this value in and then solve for x. You should get 5.7 for the length of x.
Examples using Cosine and Tangent
To solve for x Cos 28 = x/12 .8829 = x/12 x= 10.6
To solve for y Tan 42 = 12/y Note that when the variable is in the denominator it is a little more difficult to manipulate the equation to get the variable by itself. y = 12/tan 42 y = 13.3
If you have to solve for the angle given sides of a right triangle you must use the inverse sin, cos or tangent which is on the calculator as sin-1 cos-1 and tan-1
Example
cos-1 (9/15) = x x = 53 tan-1 (16/8) = y y = 63
Often trigonometric ratios are used in application problems and the angles formed in the triangles are referred to as angles of elevation and angles of depression.
The angle formed from the horizontal to the hypotenuse is called the angle of elevation if you are on the ground looking up. And the angle formed when you are above the ground and looking down with the horizontal is called the angle of depression.
The angle of elevation and the angle of depression are equal as they are alternate interior angles.
Example: If you are flying a kite and let out 80 m of string and the angle of elevation with the ground is 40 degrees and the kite string is tretched straight, how high is the kite?
Sometimes you must find a missing angle or side in non right triangle. To do this use the Law of Sines. I showed you why this worked in class but I don't have the time right now to repeat the reasons so just memorize it and I will explain in class again if you're really that interested and will put the proof on later in the week.
Law of Sines
a, b, and c are the sides opposite the angles A, B and C respectively
Example
Find the measure of <A given the measures shown.
Sin 32/ 10 = Sin <A / 12
Sin < A = 12 ( Sin 32/10)
Sine A = .6349
Sin-1 (.6359) = 39
1. Given the triangle shown find each of the following in fractional form
_________a. Sin
B
Use the Pythagorean theorem to find AB = 26 then Sin B = 10/26 or 5/13 , Cos < B 24/26 or 12/13 and the Tan of <A= 24/10 ir 12/5
_________2. Given
the triangle shown,
Drop a perpendicular from Y which divides the base in half then find the height using the Pythagorean theorem which is 8, Tan X = 8/15
3. If
RHOM is a rhombus with RO = 12 and HM = 16,
The diagonals of a rhombus bisect each other so RB = 6 and MB = 8. Use the pythag to find the side RM since the diagonals are perpendicular and form right angles. RM = 10. Set up the Sin <BRM = 8/10 or 4/5
4. Find the value of x and y to the nearest degree or unit
x=
4.
x=
5.
x=
6.
y= y= y=
# 4 ans. Set up to find x Cos x = 6/14 When you are looking for the angle make sure to use your inverse Cos-1 . x = 65 Sin y = 6/14 Sin-1 (6/14) = 25 Once you find one angle you can subtract from 90 to find the other since they are complementary
#5 ans. To find x: Tan 54 = x/12 x= 17 Cos of 54 = 12/y y= 20
#6. Sin 58 = 9/x x = 11 tan 58 = 9/y y = 6
7. Find the measure of
.
Use the law of sines Sin 35 / 12 = Sin B/15 <B = 46 degrees
8.
If the tan
A= 
, and the hypotenuse measure 39, find the lengths of the other
the other two sides of the triangle.
If the tan = 5/12 then the sides are similar to a triangle with legs of 5 and 12 and a hypotenuse of 13. Since 39 is three times greater than 39, the other two sides must be 15 and 36.
9.
The angle of depression from an observation tower to a fire is 34°. If the height
of the tower is 50 feet, how far is the fire from the tower?
Tan 34 = 50/x 74 feet
10.
A captain of a ship spots the top of a lighthouse at a 42° angle of elevation.
He
knows the house is 70 feet above the shore line.
How far is the ship from the
shore?
Tan
42 = 70/x 78 feet