math hands
math hands Trigonometry: Ch2 guide
Reference Triangles
Every angle has a reference triangle. The recipe is as follows:
Def: Inverse Cosine Function STEP 1. Choose any + r
STEP 2. Rotate θ
y
es ◦ ngl 80 ya ≤1 onl θ ◦ ≤ 0
y positive angles
2
cos
x
STEP 3. Draw a Perp to x-axis STEP 4. Label sides/signs
y
[co-domain]
Example: Find the angle θ
to estimate the sought angle. „ « 5 θ = cos−1 7
y
x
7
2
?
θ
θ
θ x
?
Def: of THE Trig Functions
≈ 0.775 (in radians [see calculator mode]... OR...) In this case, we know the adjacent ≈ 44.415◦ and the hypothenuse sides. The (in degrees [see calculator mode]) function describing such ratio is the cosine, thus
5
cos θ = ref. triangle for θ angle θ
−1
[domain] negative angles
2
r
θ
θ
r=2
ratio of sides
hyp opp
os ati 1 yr r≤ onl ≤ −1
cos
θ
5 7
Since this is not a famous ratio, we allow ourselves use of a calculator
... it should be noted that we will revisit the equation cos θ = 75 under a different context, where we will solve it completely, not limited to the domain and codomain of the arccos function.
adj
sin θ =
opp hyp
cos θ =
adj hyp
tan θ =
opp adj
csc θ =
hyp opp
sec θ =
hyp adj
cot θ =
adj opp
Def: Inverse Tangent Function LOP Diagram: ◦ es ngl 90 ya θ< onl ◦ < 0 −9
l rea al l ios rat
tan
• The essential concept 1: once the ratios are known, one side is enough to determine the other sides
r
θ
• The essential concept 2: For each angle, the ratios are described by the trig functions where defined.
−1
tan [domain]
[co-domain]
Def: Inverse Sine Function ◦ es ngl 90 ya θ≤ onl ◦ ≤ 0 −9
sin
p hy
r
θ
sin [domain]
os ati yr ≤1 onl ≤r 1 −
× sin θ os θ ×c
−1
[co-domain]
θ
math hands
×
opp
nθ ta
adj
c
2007-2009 MathHands.com
Solving Euclidean Right Triangles Solve the triangle [may not be drawn to scale assume typical units for length]:
β 23
34◦
b
a
Solution: Notice, this proposition is typical, three items are given; the right angle, the 34 angle, and the hypothenuse side, while three items are missing; sides a, b, and angle β. It should be noted that the angle β is relatively easy to determine β + 34◦ = 90◦ Therefore, β = 56◦ . Now, we solve for b, the side opposite of the 34◦ angle. Since we know the hypothenuse is 23 units, and we want to know the sine function describes this ratio, sin 34◦ ≈ 0.559, similarly to solve for a we use the cosine ratio, cos 34◦ ≈ 0.829. Thus, we illustrate the ratios on the triangle:
56◦ 23
×0.559
12.861
×0.829
34◦
19.068
math hands
c
2007-2009 MathHands.com
Reference Triangles
Every angle has a reference triangle. The recipe is as follows:
Def: Inverse Cosine Function STEP 1. Choose any + r
STEP 2. Rotate θ
y
es ◦ ngl 80 ya ≤1 onl θ ◦ ≤ 0
y positive angles
2
cos
x
STEP 3. Draw a Perp to x-axis STEP 4. Label sides/signs
y
[co-domain]
Example: Find the angle θ
to estimate the sought angle. „ « 5 θ = cos−1 7
y
x
7
2
?
θ
θ
θ x
?
Def: of THE Trig Functions
≈ 0.775 (in radians [see calculator mode]... OR...) In this case, we know the adjacent ≈ 44.415◦ and the hypothenuse sides. The (in degrees [see calculator mode]) function describing such ratio is the cosine, thus
5
cos θ = ref. triangle for θ angle θ
−1
[domain] negative angles
2
r
θ
θ
r=2
ratio of sides
hyp opp
os ati 1 yr r≤ onl ≤ −1
cos
θ
5 7
Since this is not a famous ratio, we allow ourselves use of a calculator
... it should be noted that we will revisit the equation cos θ = 75 under a different context, where we will solve it completely, not limited to the domain and codomain of the arccos function.
adj
sin θ =
opp hyp
cos θ =
adj hyp
tan θ =
opp adj
csc θ =
hyp opp
sec θ =
hyp adj
cot θ =
adj opp
Def: Inverse Tangent Function LOP Diagram: ◦ es ngl 90 ya θ< onl ◦ < 0 −9
l rea al l ios rat
tan
• The essential concept 1: once the ratios are known, one side is enough to determine the other sides
r
θ
• The essential concept 2: For each angle, the ratios are described by the trig functions where defined.
−1
tan [domain]
[co-domain]
Def: Inverse Sine Function ◦ es ngl 90 ya θ≤ onl ◦ ≤ 0 −9
sin
p hy
r
θ
sin [domain]
os ati yr ≤1 onl ≤r 1 −
× sin θ os θ ×c
−1
[co-domain]
θ
math hands
×
opp
nθ ta
adj
c
2007-2009 MathHands.com
Solving Euclidean Right Triangles Solve the triangle [may not be drawn to scale assume typical units for length]:
β 23
34◦
b
a
Solution: Notice, this proposition is typical, three items are given; the right angle, the 34 angle, and the hypothenuse side, while three items are missing; sides a, b, and angle β. It should be noted that the angle β is relatively easy to determine β + 34◦ = 90◦ Therefore, β = 56◦ . Now, we solve for b, the side opposite of the 34◦ angle. Since we know the hypothenuse is 23 units, and we want to know the sine function describes this ratio, sin 34◦ ≈ 0.559, similarly to solve for a we use the cosine ratio, cos 34◦ ≈ 0.829. Thus, we illustrate the ratios on the triangle:
56◦ 23
×0.559
12.861
×0.829
34◦
19.068
math hands
c
2007-2009 MathHands.com
