Download notes 4_3 right triangle trig

4.3 Right Triangle Trigonometry
As derived from the Greek language, the word trigonometry means “measurement of
triangles.” Initially, trigonometry dealt with the relationships among the sides and
angles of triangles and was used in the development of astronomy, navigation, and
surveying. With the development of calculus and physical sciences in the 17th century, a
different perspective rose – one that viewed the classic trig relationships as functions
with the set of real numbers as their domains. This text incorporates both those
perspectives, but we will deal with the triangle view first.
These are all related to right triangles. Relative to the angle 𝜽 (“theta”), the 3 sides of a
triangle are the hypotenuse, the opposite side, and the adjacent side.
Using these sides, you can form six ratios that define the 6 trig functions of the acute
angle 𝜃.
Sine (sin)=
Cosecant (csc) =
Cosine (cos) =
Secant (sec) =
Tangent (tan) =
Cotangent (cot) =
Ex 1: Use the triangle to find the values of the 6 trig functions of 𝜃.
3
4
There are some special angles that we frequently use in trig. We will now find the trig
values of these angles.
The 450 – 450 – 900 Triangle:
1
1
The 300 – 600 – 900 Triangle:
2
If we need to answer questions that involve these angles, we usually use the measures
above, leaving them in square root form.
Ex 2: Use the given values of sin 600 =
a. Tan 600
√3
2
and cos 600 = ½ to find the following:
b. Sin 300
Ex 3: A surveyor is standing 50 feet from the base of a large tree. The surveyor
measures the angle of elevation to the top of the tree as 71.50. How tall is the tree?
Ex 4: A person is 200 yards away from a river. Rather than walk directly to the river, the
person walks 400 yards along a straight path to the river’s edge. Find the acute angle 𝜃
between this path and the river’s edge.
Ex 5: Specifications for a loading dock ramp require a rise of 1 foot for each 3 feet of
horizontal length. In the figure below, find the lengths of sides b and c and find the
measure of 𝜃.
C
b
4 ft