Trig Test Notes
Sine Law & Cosine Law
Sine Law
When do we use Sine Law?
to determine the side lengths or angles of a triangle when you have specific combinations of known information:
● 2 sides and 1 non-contained (meaning one of the sides that make up the angle is unknown) angle/angle opposite to 1 of
the sides (SSA)
● 2 angles and 1 side that is part of at least 1 side of the angle (AAS or ASA)
Formula:
● sin×A/a=sin×B/b=sin×C/c
● sin×a/A=sin×b/B=sin×c/C
Example:
A surveyor is standing at point A and measures the angle between two landmarks B and C to be 40°. The surveyor then walks 300
meters to point B, where the angle between points A and C is measured to be 65°. How far is point C from point A?
(B=65°)
(SIDE
1. Find missing angle
(Remember, sum of angles of triangle=180°)
180°-40°-65°=75°
C=75°
2. Apply Sine Law
c/sin(65°)=300/sin(75°)
3. Solve for c
c=300×sin(65°)/sin(75°)
(calculate sins)
sin(65°)=0.9063
sin(75°)=0.9659
(plug values)
c=300×0.9063/0.9659
=271.89/0.9659
=281.4
4. Final Answer
c=281.4m
Therefore, the distance between from pointA-pointC is approximately 281.4m.
Cosine Law
When do we use Cosine Law?
● two sides and the included angle between those 2 known sides (SAS)
● when you know all three sides of a triangle (SSS) and need to find a missing angle
Formula:
● a²=b²+c²- 2bc×cosA
● b²=c²+a²-2ca×cosB
● c²=a²+b²-2ab×cosC
Example:
, 1. Use Cosine Law
● a=12
● b=16
● C=120°
(using Cosine Law)
c²=a²+b²-2abcosC =>
c²=12²+16²-2×12×16cos120°
2. Calculate
12²=144
16²=256
2×12×16=384
cos120°=-0.5
(plug values)
c²=144+256-384×-0.5
c²=400+192=592
c=√592
=24.3
3. Final Answer
c=24.3m
Therefore, the straight-line distance from pointA-C is approximately 24.3m.
Test review answers
1. Find values for each of the following. (Answers must be fractions where possible, otherwise to 2
decimal places. (right angled triangles)
a.
(remember SOHCAHTOA)
cos𝜃=ada/hyp =>
24/25
1. Use cos⁻¹
𝜃=cos⁻¹(24/25)
𝜃=cos⁻¹×0.96
𝜃=16.26°
2. Final Answer
𝜃=16.26°
tan𝜃=opp/ada =>
1. Use tan⁻¹
𝜃=tan⁻¹(7/24)
𝜃=tan⁻¹×0.2917
𝜃=16.26°
2. Final Answer
Sine Law & Cosine Law
Sine Law
When do we use Sine Law?
to determine the side lengths or angles of a triangle when you have specific combinations of known information:
● 2 sides and 1 non-contained (meaning one of the sides that make up the angle is unknown) angle/angle opposite to 1 of
the sides (SSA)
● 2 angles and 1 side that is part of at least 1 side of the angle (AAS or ASA)
Formula:
● sin×A/a=sin×B/b=sin×C/c
● sin×a/A=sin×b/B=sin×c/C
Example:
A surveyor is standing at point A and measures the angle between two landmarks B and C to be 40°. The surveyor then walks 300
meters to point B, where the angle between points A and C is measured to be 65°. How far is point C from point A?
(B=65°)
(SIDE
1. Find missing angle
(Remember, sum of angles of triangle=180°)
180°-40°-65°=75°
C=75°
2. Apply Sine Law
c/sin(65°)=300/sin(75°)
3. Solve for c
c=300×sin(65°)/sin(75°)
(calculate sins)
sin(65°)=0.9063
sin(75°)=0.9659
(plug values)
c=300×0.9063/0.9659
=271.89/0.9659
=281.4
4. Final Answer
c=281.4m
Therefore, the distance between from pointA-pointC is approximately 281.4m.
Cosine Law
When do we use Cosine Law?
● two sides and the included angle between those 2 known sides (SAS)
● when you know all three sides of a triangle (SSS) and need to find a missing angle
Formula:
● a²=b²+c²- 2bc×cosA
● b²=c²+a²-2ca×cosB
● c²=a²+b²-2ab×cosC
Example:
, 1. Use Cosine Law
● a=12
● b=16
● C=120°
(using Cosine Law)
c²=a²+b²-2abcosC =>
c²=12²+16²-2×12×16cos120°
2. Calculate
12²=144
16²=256
2×12×16=384
cos120°=-0.5
(plug values)
c²=144+256-384×-0.5
c²=400+192=592
c=√592
=24.3
3. Final Answer
c=24.3m
Therefore, the straight-line distance from pointA-C is approximately 24.3m.
Test review answers
1. Find values for each of the following. (Answers must be fractions where possible, otherwise to 2
decimal places. (right angled triangles)
a.
(remember SOHCAHTOA)
cos𝜃=ada/hyp =>
24/25
1. Use cos⁻¹
𝜃=cos⁻¹(24/25)
𝜃=cos⁻¹×0.96
𝜃=16.26°
2. Final Answer
𝜃=16.26°
tan𝜃=opp/ada =>
1. Use tan⁻¹
𝜃=tan⁻¹(7/24)
𝜃=tan⁻¹×0.2917
𝜃=16.26°
2. Final Answer
