5.1 Trigonometric Ratios of Acute Angles.notebook

5.1 Trigonometric Ratios of Acute Angles.notebook
April 02, 2015
2015 04 02
5.1 Trigonometric Ratios of Acute Angles
The Pythagoran Theorem
The Pythagorean Theorem is an equation which describes the relationship between the 3 sides of a right angle triangle.
a
c
b
c2=a2+b2 , where 'c' represents the hypotenuse which is the longest side and the side that is opposite the right angle.
Sides 'a' and 'b' are interchangeable, so long as side 'c' is the hypotenuse.
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5.1 Trigonometric Ratios of Acute Angles.notebook
April 02, 2015
SOHCAHTOA is an acronym that we use to help remember the primary trigonometric ratios. These ratios are used to examine the relationship between the sides of a right angle triangle and its interior angles.
H
θ ‐ theta.
O
θ
A
The hypotenuse is always assigned the letter H. The other two sides are labeled according to their position relative to the angle θ. The side opposite the angle is labeled 'O' for opposite and the other side which is used to form angle θ with the hypotenuse is designated 'A' for adjacent.
A
θ
H
O
When the angle we are interested in changes, the hypotenuse will remain the same, but the other two sides are switched.
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5.1 Trigonometric Ratios of Acute Angles.notebook
April 02, 2015
Ex. Solve for x in each of the following triangles.
A
12cm
C
x
16cm
B
A
x
C
25m
7m
B
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5.1 Trigonometric Ratios of Acute Angles.notebook
April 02, 2015
Solving for a Side
In order to solve for a side in a right angle triangle, set up the appropriate trig ratio, isolate the variable of interest and solve.
41cm
35o
x
31o
64m
x
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5.1 Trigonometric Ratios of Acute Angles.notebook
April 02, 2015
Solving for an Angle
To solve for an angle, it will be necessary to take the inverse of the primary trig. ratio (ex. sin ). Identify the two sides you are given, set up the appropriate ratio and solve for θ by taking the inverse of the trig ratio.
‐1
18cm
θ
13cm
Solving a Triangle
When you are asked to "Solve a Triangle" this means you are required to find the value of all unknown sides and angles.
As much as possible, use given angles rather than calculated ones.
Ex. Solve the following triangle
14cm
θ
24cm
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5.1 Trigonometric Ratios of Acute Angles.notebook
April 02, 2015
Secondary Trig. Ratios
Along with the primary trig ratios sine, cosine and tangent, there also exist secondary ratios, cosecant, secant and cotangent, which are the reciprocals of each of the primary trig. ratios.
secant
cosecant
cotangent
Ex. Determine each of the primary and secondary trig ratios for the following triangles. Then determine the value of θ.
14cm
50cm
θ
48cm
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5.1 Trigonometric Ratios of Acute Angles.notebook
April 02, 2015
Determine the value of the following expressions.
(3 decimal places)
a) sin 35o
b) sec 42o
Determine the value of θ in each of the following expressions.
a) csc θ = 3.4
b) cot θ = 0.67
Pg. 280 #1‐7,11,14,18
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