Download Keeper 10 - Right Triangle Trig

GPS Pre-Calculus
Keeper 10
Trigonometric Ratios
The lengths of the sides of a right triangle are used
to define the six trigonometric ratios:
sine (sin)
cosine (cos)
tangent (tan)
cosecant (csc)
secant (sec)
cotangent (cot)
Hypotenuse

Side adjacent to 
Side opposite 
Trigonometric Function Values of an
Acute Angle 
Let  be an acute angle of a right triangle. Then the
six trigonometric functions of  are as follows:
side opposite 
sin 
hypotenuse
hypotenuse
csc 
side opposite 
side adjacent to 
cos 
hypotenuse
hypotenuse
sec 
side adjacent to 
side opposite 
tan 
side adjacent to 
side adjacent to 
cot  
side opposite 
Memorizing Trigonometric Ratios
The mnemonic device
SOH – CAH – TOA
Is most commonly used to remember the ratios
for sine, cosine, and tangent.
Reciprocal Functions
Note that there is a reciprocal relationship
between pairs of the trigonometric functions.
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
Common Relationships
From the definitions of the sine, cosine, tangent,
and cotangent functions, you can also derive the
following relationships.
Example
In the triangle shown, find the six trigonometric
function values of (a)  and (b) .

12
13

5
Example
6
If sin  
and  is an acute angle, find the other
7
five trigonometric function values of .
Find the value of x. Round to the
nearest tenth.
Application
A competitor in a triathlon is running along the
course shown. Determine the length in feet that
the runner must cover to reach the finish line.
Investigation of Special Right
Triangle
Trigonometric Values of Special Angles
Inverse Trigonometric Functions
Find a Missing Angle Measure
Angle of Elevation and Depression
Some applications of trig use an angle of elevation
or depression.
An angle of elevation is the angle formed by a
horizontal line and an observer’s line of sight to an
object above.
An angle of depression is the angle formed by a
horizontal line and an observer’s line of sight to an
object below.
Using an Angle of Elevation
A ground crew worker who is 6 feet tall is
directing a plane on a runway. If the worker
sights the plane at an angle of elevation of 32
degrees, what is the horizontal distance from
the worker to the plane?
Application
A hot air balloon that is moving above a
neighborhood has an angle of depression of 28
degrees to one hose and 52 degrees to a house
down the street. If the height of the balloon is
650 feet, estimate the distance between the two
houses.
Solving Right Triangles
Trigonometric functions and inverse relations
can be used to solve a right triangle, which
means to find the measures of all of the sides
and angles of the triangle.