Download right triangle trigonometry – section 4.3

RIGHT TRIANGLE TRIGONOMETRY – SECTION 4.3
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of
triangles. Right Triangle Trigonometry deals with the ratios of the lengths of the sides of the triangles.
One of the first accomplishments of Trigonometry was comparing the distance between Earth and the
moon with the distance between Earth and the sun using ratios. This discovery constituted the formal
beginning of Trigonometry (about 140 BC).
The six Right Triangle Trigonometric Functions and their definitions:
The 1st trig function is called the Sine and it is abbreviated as “sin”. Sine is defined as the ratio of the
opposite side of an angle and the hypotenuse. This ratio depends on the angle; therefore, the angle must
be stated always after the function (sin). Ex: sinA, sinβ, etc. The word Sine or sin by itself has no meaning.
Name of Trig
Function
Sine
Abbreviation
Definition
sin
Cosine
cos
Tangent
tan
Cotangent
cot
Secant
sec
Cosecant
csc
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑜𝑓 𝜃
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝑜𝑓 𝜃
cos 𝜃 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑜𝑓 𝜃
tan 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝑜𝑓 𝜃
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝑜𝑓 𝜃
cot 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑜𝑓 𝜃
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sec 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝑜𝑓 𝜃
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
csc 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑜𝑓 𝜃
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
sin 𝜃 =
SOHCAHTOA
Reciprocal Trig Functions
Notice that the following trig functions are reciprocals of each other:
 sin Ѳ and csc Ѳ
 cos Ѳ and sec Ѳ
 tan Ѳ and cot Ѳ
Ex:1
Ex:2
𝜃
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
𝜃
Function values of Special Angles:
Recall from Geometry the special triangles 30-60-90 an 45-45-90.
30-60-90
45-45-90
Relationships:
hypotenuse = 2(short side)
long side = (short side)√3
hypotenuse = (side) √2
Fill in the table below using the special triangles:
30ᵒ
60ᵒ
45ᵒ
sin
cos
tan
Co-Functions
Recall the definition of Complementary Angles: Two angles whose sum = 90ᵒ.
The following trig functions are co-functions:
 sine and cosine
 tangent and cotangent
 secant and cosecant
Notice co-functions use the same trig term with the prefix “co”.
If two trig functions are co-functions and their angles are complementary, then their values are equal.
Ex: sin 10ᵒ = cos 80ᵒ
Write 3 more examples here:
or tan 22ᵒ = cot 68ᵒ
or csc 34ᵒ = sec 56ᵒ
Applications of Right Triangle Trigonometry
Right Triangle Trig can be used to solve word problems which cannot be solved with Algebra and
Geometry alone.
Angle of Elevation: An angle formed by a horizontal line and the line of sight to an object above the
horizontal line .(Think of an airplane taking off).
Drawing:
Angle of Depression: An angle formed by a horizontal line and the line of sight to an object below the
horizontal line .(Think of an airplane landing)
Drawing:
Both the angle of elevation and depression are always acute angles.
Ex:6
Ex:7