Geometry Day 58

1
Geometry Day 58

• Trigonometry

2
Todays Objective

• Right Triangle Trigonometry
• Identify and apply the three basic trigonometric
  ratios
• Sine
• Cosine
• Tangent
• Identify the inverse trig functions

3
What is Trigonometry?

• Trigonometry is a branch of mathematics that
  involves finding out information about right
  triangles.
• (Trigonometry can also involve circles, but that
  is a subject for Pre-Calculus.)
• To do trigonometry, you will need a scientific or
  graphing calculator.
• Make sure your calculator is in degree mode.

4
Introduction to Trig functions

• Your handout contains a table with columns
  labeled Sine, Cosine, and Tangent. These are
  often abbreviated as Sin, Cos, and Tan (but they
  are pronounced the same way).
• Locate these buttons on your calculator.
• Dont worry about what these mean right now, but
  use your calculator to confirm the values in the
  table.
• In other words, check sin 0 and sin 1 (for
  example) on your calculator and see if it matches
  whats on the table.
• Now use your calculator to fill in the missing
  values on the table.
• Also, use your calculator to add two rows 89.5
  and 90.
• While doing so, see if you can make any
  observations about the numbers on the table.

5
Trig and Similarity
hyp
opp

• Consider a right triangle.
• Lets say we know one of theacute angles.
• Any right triangle we draw with an angle of x
  degrees will be similar to this triangle. Why?
• Since all triangles with this set of angles are
  similar, then their sides will always be in the
  same ratio.
• If Im standing at angle x, then I can label the
  three sides as follows
• There is the leg across from me, which Ill call
  the opposite leg.
• The leg next to me will be the adjacent leg.
• Well call the hypotenuse, the hypotenuse.

x
adj
6
Trig and Similarity
hyp

• Note that these side labelsare from the
  perspective of theangle were working with. If
  wewere standing at the other acuteangle, the
  opposite and adjacent sides would switch. (The
  hypotenuse will always be the hypotenuse.)
• Remember, any right triangle with an angle
  measuring x degrees will be similar, and
  therefore will have proportional sides.
• So, if I were to take two of these sides and form
  a ratio, that ratio will be consistent, no matter
  how big or small the triangle.

opp
x
adj
7
Trig and Similarity
hyp

• Ancient mathematicians noticedthis, and
  calculated what the ratiosof the sides are for
  the differentpossible acute angles (e.g., 10?,
  20?,36?, etc.).
• Sine, cosine, and tangent refer to the ratios of
  specific pairs of sides

opp
x
adj
8
Trig and Similarity
hyp

• So sin 40? .6428 means that, whenever a right
  triangle has a 40? angle, the ratio of the
  opposite side to the hypotenuse is .6428.
• In the past, these values were written down in
  tables, and if you needed them, you would have to
  look them up. Today, these tables are programmed
  into your calculator.

opp
x
adj
9
Sohcahtoa

• Many students have a hard time remembering the
  ratios. The word sohcahtoa can help.
• Some old hippie caught another hippie trippin on
  acid.
• Feel free to create your own pneumonic device.

10
Observations

• Sine and cosine will always be between zero and
  one. Why?
• Tangent starts very small and grows very large.
  Why?
• What is sin A?
• What is cos B?
• How are ?A and ?B related?
• Tangent is opp/adj. Another definition is
  sin/cos.

A
B
C
11
Application

• How is this useful?
• Solve for x and y.
• Ask
• From the perspective of this angle,which sides
  am I working with?
• Which trig function relates those sides?
• To prevent rounding errors, solve the equation
  for the variable before you touch your calculator!

x
10
y
25?
12
Trig and Special Right Triangles

• We know the relationships among sides of two
  right triangles
• Given this knowledge, what are the following
  values?

1
30?
45?
2
1
?3
?2
60?
1
13
Reciprocal Trig Functions
A

• There are three other trig functions,but they
  are used less. (In fact, itisnt necessary to
  use them at all!)
• Cosecant (or csc)
• Secant (or sec)
• Cotangent (or cot)
• In other words, these are the multiplicative
  reciprocals of sin, cos, and tan, respectively.

B
C
14
Reciprocal Trig Functions
A

• Lets say we wanted to solve for x.
• Either of these ratios would beappropriate
• Most calculators dont have the reciprocal
  functions, so its probably best to use cosine.

28?
x
18
B
C
15
Inverse Trig Functions
A

• What if we know the sides of a righttriangle,
  and need to find an angle?
• Every mathematical function hasan inverse. The
  inverse undoeswhat the function did.
• The inverses of sin, cos, and tan, respectively,
  are sin-1, cos-1, and tan-1.
• These are sometimes written as arcsin, arccos,
  and arctan.
• Remember, sin takes an angle and gives a ratio.
  Sin-1 takes a ratio and gives us the angle.

5
3
B
C
16
Inverse Trig Functions
A

• We would set up the equation inthe same way.
• What equation would we set upto solve for ?A?
• To solve for the angle, we use the inverse on the
  ratio.
• Solve for ?B using this technique.

5
3
B
C
17
Solving a Right Triangle
A

• To solve a right triangle is to findall the
  measurements of its sidesand angles.
• You can solve a right triangle if youknow
• Two side lengths, or
• One side length and the measurement of one acute
  angle.
• Solve this right triangle.

2
18?
B
C
18
Solving a Right Triangle
A

• Solve this right triangle.

19
12
B
C
19
Homework 34

• Workbook pp. 103-104