Chapter 5
Trigonometric Functions
Section 5.2
Trigonometric Functions of Acute Angles
Trigonometric Functions of Acute Angles
When working with right triangles, it is
convenient to refer to the side opposite an angle
or the side adjacent to (next to) an angle.
Trigonometric Functions of Acute Angles
Consider an angle q in the right triangle shown
below. Let x and y represent the lengths,
respectively, of the adjacent and opposite side
of the triangle, and let r be the length of the
hypotenuse. Six possible ratios can be formed:
𝑦
𝑟
𝑥
𝑟
𝑦
𝑥
𝑟
𝑦
𝑟
𝑥
𝑥
𝑦
Trigonometric Functions of Acute Angles
sin q =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos q =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
tan q =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
csc q =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
sec q =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
cot q =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
Trigonometric Functions of Acute Angles
Example 1
Find the six trigonometric functions of q for the
triangle given in the Figure 5.32 below.
Example 2
Given that q is and acute angle and
5
8
cos q = , find tan q.
Trigonometric Functions of Special Angles
Special Angles
Trigonometric Functions of Special Angles
Example
Find the exact value of sin2450 + cos2600.
Reciprocal Functions
sin q =
1
csc θ
tan q =
1
cot θ
csc q =
1
sin θ
cos q =
1
sec θ
sec q =
1
cos θ
cot q =
1
tan θ
Applications
From a point 115 feet from the base of a
redwood tree, the angle of elevation to the top
of the tree is 64.30. Find the height of the tree to
the nearest foot.
Applications
If the distance from a plane to a radar station is
160 miles and the angle of depression is 330,
find the number of ground miles from a point
directly below the plan to the radar station.
Applications
The angle of elevation from point A to the top of
a space shuttle is 27.20. From a point 17.5
meters further from the space shuttle, the angle
of elevation is 23.90. Find the height of the
space shuttle.
Assignment
Section 5.2 - Worksheet