4.11.4 Trigonometry
The student is able to (I can):
For any right triangle
Define the sine, cosine, and tangent ratios and their Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
By the Angle-Angle Similarity Theorem, a right triangle with a given acute angle is similar to every other right triangle with the same acute angle measure. This means that the ratios between the sides of those triangles are always the same.
Because these ratios are so useful, they were given names: sinesinesinesine, cosinecosinecosinecosine, and were given names: sinesinesinesine, cosinecosinecosinecosine, and tangenttangenttangenttangent. These ratios are used in the study of trigonometry.
sine sine of A
AAAA
hypotenuse
adjacent
opposite
leg opposite AsinA
hypotenuse
= =
cosine
tangent
cosine of A
tangent of A
hypotenuse
leg adjacent to AcosA
hypotenuse
= =
leg opposite AtanA
leg adjacent to A
= =
We can use the trig ratios to find either missing sides or missing angles of right triangles. To do this, we will set up equations and solve for the missing part. In order to figure out the sine, cosine, and tangent ratios, we can use either a calculator or a trig table.
To use the Nspire calculator to find tan 51:
From a New Document, press the key:
Use the right arrow key ( ) to select tan Use the right arrow key () to select tan and press :
Type 5I and hit :
To use the calculator on your phone:
Turn your phone landscape to access the scientific calculator.
Type the angle in first, thenthenthenthen select tan.
To use a trig table to find cos 52:
Locate 52 on the table. Scan over to the Cos column and find
the value.
cos 52 = .6157
To find an angle, we use the inverseinverseinverseinverse trig functions (in more advanced classes, you will hear them referred to as arcsine, arccosine, and arctangent). On your calculator, these are listed as sin1, cos1, and tan1.
Ex. Find :
1 8sin17
Press the button, and then the arrow to select sin1. Then enter 8p17. You should get 28.07
This means that the angle opposite a leg of 8 with a hypotenuse of 17 will measure around 28.
17
To find an angle using a trig table, just find the appropriate trig column, find the closest value, and read back to the angle.
Ex. Find ( )1tan 0.35
0.35 is closer to 0.3443 than it is to 0.3640, so our answer would be 19.
You will be expected to memorize these ratio relationships. There are many hints out there to help you keep them straight. The most common is SOHSOHSOHSOH----CAHCAHCAHCAH----TOATOATOATOA , where
A mnemonic I like is Some Old Hippie
pOS
pin
pHy=
dAC
jos
pHy=
pOT
pan
jAd=
A mnemonic I like is Some Old Hippie Caught Another Hippie Trippin On Acid.
Or Silly Old Hitler Couldnt Advance His Troops Over Africa.
Examples I. Use the triangle to find the following ratios.
1. sin A = _____
A
BC
8
15
17
1. sin A = _____
2. cos A = _____
3. tan A = _____
Examples I. Use the triangle to find the following ratios.
1. sin A = _____
A
BC
8
15
17
15
171. sin A = _____
2. cos A = _____
3. tan A = _____
8
17
17
15
8
Examples I. Use the triangle to find the following ratios.
4. sin B = _____
A
BC
8
15
17
8
174. sin B = _____
5. cos B = _____
6. tan B = _____
17
15
17
8
15
Examples II. Find the lengths of the sides to the nearest tenth.
1. x (opp)
15(adj)
58
=
=
xtan58
15x 15tan58
24.0
2. 26
(hyp)
x(adj)
46
=
=
xcos46
26x 26cos46
18.1
III. Find the missing angle to the nearest whole degree.
26 (hyp)
19 (opp)
x
=19
sinx26
=
1 19x sin26
x 47
