Download Chapter 2 Acute Angles and Right Triangles

Chapter 2
Section 2 – 1
Acute Angles and Right Triangles
Trigonometric Functions of Acute Angles
Right Triangle – Based Definitions of Trigonometric Functions
For any acute angle A in standard position,
sin A 
y Side Opposite

r
Hypotenuse
csc A 
r
Hypotenuse

y Side Opposite
cos A 
x Side Adjacent

r
Hypotenuse
sec A 
r
Hypotenuse

x Side Adjacent
tan A 
y Side Opposite

x Side Adjacent
cot A 
x Side Adjacent

y Side Opposite
SohCahToa
Cofunction Identities
For any acute angle A,
sin A  cos  90o  A 
csc A  sec  90 o  A 
tan A  cot  90 o  A 
cos A  sin  90o  A 
sec A  csc  90 o  A 
cot A  tan  90 o  A 
Functions Values of Special Angles
θ
sin θ
cos θ
tan θ
o
1
30
3
3
2
o
45
60o
2
2
3
2
2
2
2
1
2
cot θ
sec θ
3
2 3
3
2
3
1
1
3
3
3
2
csc θ
2
2
2 3
3
Section 2 – 2
Trigonometric Functions of Non – Acute Angles
Reference Angle – the positive acute angle made by the terminal side of
Angle θ and the x – axis.
Note: The reference angle must always be found from the x – axis!
Finding Trigonometric Function Values for Any Nonquadrantal Angle
Step 1 – If   360o , or if   0o , find a coterminal angle by adding or
subtracting 360o as many times as needed to an angle 0o    360o
Step 2 – Find the reference angle  '
Step 3 – Find the necessary values of the trigonometric functions for the
reference angle  '
Step 4 – Determine if the correct signs for the values found in Step 3. Use
the table in your notes or remember that All Students Take Calculus
Section 2 – 3
Calculator
Finding Trigonometric Function Values Using a
Degree Mode Vs. Radian Mode
To find the sec, csc, and cot.
1. Find the sin, cos, or tan of the desired angle
2. Use the x -1 button to find its reciprocal
To find θ, the angle, use the sin -1, cos -1, or tan -1 buttons
To find the csc -1, sec -1 or the cot -1, do the following,
1. Call up the reciprocal of the function you need.
2. Then take
1
value

3. It should look like this on you calculator sin 1 

 3.1245 
1
Section 2 – 4 Solving Right Triangles
Exact Number – a number that represents the result of counting, or a number
that results from theoretical work and is not the result of a measurement
Solving a Triangle – find the measures of all the angles and sides of the
triangle.
Angle of Elevation – the angle measured from the horizontal up
Angle of Depression – the angle measured from the horizontal down
Solving an Applied Trigonometry Problem
Step 1 – Draw a sketch, and label it with the given information. Label the
quantity to be found with a variable.
Step 2 – Use the sketch to write up an equation relating the given quantities
to the variable.
Step 3 – Solve the equation, and check that your answer makes sense.
Section 2 – 5
Further Applications of Right Triangles
Bearing – used in navigation. Two methods are used to solve problems
involving bearings.
Method 1 – when a single number is given as a bearing, it is understood that
the bearing is measured in a clockwise direction from due north.
Method 2 – when a bearing is given with a direction, then angle, then
direction, (N 45o E) the bearing is measured from the first direction (Due
North) in the rotation of the second direction East.