Download Section 5.6 - TopCatMath

Math 170 - Cooley
Pre-Calculus
OCC
Section 5.6 – Right Triangle Trigonometry
Theorem – Trigonometric Ratios
If (x, y) is any point other than the origin on the terminal side of an angle α in standard position and r  x 2  y 2 ,
then,
y
x
y
sin   , cos   , and tan    x  0 .
r
r
x
Theorem – Trigonometric Functions of an Acute Angle of a Right Triangle
If  is an acute angle of a right triangle, then
sin  
opp
,
hyp
cos  
adj
,
hyp
and
tan  
opp
.
adj
Strategy: Solving a Right Triangle
1. Use the Pythagorean theorem to find the length of a third side when the lengths of two sides are known.
2. Use the trigonometric ratios to find missing sides or angles.
3. Use the fact that the sum of the measures of the angles of a triangle is 180o to determine a third angle
when two are known.
Applications involving Angle of Elevation & Angle of Depression
Using trigonometry, we can find the size of an object without actually measuring the object but by measuring an
angle.
Two common terms used in this regard are angle of elevation and angle of depression.

The angle of elevation  for a point above a horizontal line is the angle formed by the horizontal line
and the observer’s line of sight through the point.

The angle of depression  for a point below a horizontal line is the angle formed by the horizontal line
and the observer’s line of sight through the point.
 Exercises:
Find exact values for sin  , cos , tan  , sin  , cos  , and tan  for the given right triangle.
1)
2)
-1-
Math 170 - Cooley
Pre-Calculus
OCC
Section 5.6 – Right Triangle Trigonometry
 Exercises:
Assume that  is an angle in standard position whose terminal side contains the given point and that
0    90 . Find the degree measure of  to the nearest tenth of a degree.
3) (4, 5)
4) (4.3, 6.9)
Assume that  is an angle in standard position whose terminal side contains the given point and that
0  

2
. Find the degree measure of  to the nearest tenth of a degree.
1 1
5)  , 
3 2
6) ( 7,
3)
Solve each right triangle with the given sides and angles. In each case, make a sketch. Note that  is the acute
angle opposite leg a and  is the acute angle opposite leg b. the hypotenuse is c.
7)   45, c  10
8)   47, a  3
-2-
Math 170 - Cooley
Pre-Calculus
OCC
Section 5.6 – Right Triangle Trigonometry
 Exercises: Solve each problem.
9)
A hiker stands 80 feet from a giant redwood tree and sights the top with an angle of elevation of 75 .
How tall is the tree to the nearest foot?
10)
To get from Muleshoe to Snyder, Harry drives 50 mph for 178 mi south on route 214 to Seminole, then
goes east on route 180 to Snyder. Harriet leaves Muleshoe one hour later at 55 mph, but takes US 84,
which goes straight from Muleshoe to Snyder through Lubbock. If US 84 intersects route 180 at a
50 angle, then how many more miles does Harry drive?
11)
A forest ranger atop a 3248-ft mesa is watching the progress of a forest fire spreading in her direction. In
5 minutes the angle of depression of the leading edge of the fire changed from 11.34 to 13.51 . At what
speed in miles per hour is the fire spreading in the direction of the ranger?
12)
For years the Woolworth skyscraper in New York held the record for the world’s tallest office building. If
the length of the shadow of the Woolworth building increases by 17.4 m as the angle of elevation of the
sun changes from 44 to 42 , what is the height of the building?
-3-