TRIGONOMETRY FORMULA
Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles.
Use sine,cosine or tangent to find the value of side x in the triangle below.
Step 1) Based on your givens and unknowns, determine which sohcahtoa ratio to use Since we know the 53 angle, the hypotenuse,and we want to find length of the opposite side, we should use sine
Step 2) Set up an equation based on the ratio you chose in the step 1 sin(53)=oppositehypotenuse
sin(53)=x15
Step 3) Cross multiply and solve the equation for the side length. (round to the nearest hundredth) 15×sin(53)=x
x≈11.98
Use sine,cosine or tangent to find the value of side x in the triangle below.
Step 1) Based on your givens and unknowns, determine which sohcahtoa ratio to use Since we know the 53 angle, the hypotenuse,and we want to find length of the opposite side, we should use sine
Step 2) Set up an equation based on the ratio you chose in the step 1 sin(53)=oppositehypotenuse
sin(53)=x15
Step 3) Cross multiply and solve the equation for the side length. (round to the nearest hundredth) 15×sin(53)=x
x≈11.98
The Law of Sine and Cosine
Sine
a/sin A=b/sin B=c/sin C
So if you divide side a by the sine of angle A it is equal to side b divided by the sine of angle B, and also equal to side c divided by the sine of angle C
Example:
So if you divide side a by the sine of angle A it is equal to side b divided by the sine of angle B, and also equal to side c divided by the sine of angle C
Example:
Law of Sines: a/sin A = b/sin B = c/sin C
Put in the values we know: a/sin A = 7/sin(35°) = c/sin(105°)
Ignore a/sin A (not useful to us): 7/sin(35°) = c/sin(105°)
Now we use our algebra skills to rearrange and solve:
Swap sides: c/sin(105°) = 7/sin(35°)
Multiply both sides by sin(105°): c = ( 7 / sin(35°) ) × sin(105°)
Calculate: c = ( 7 / 0.574... ) × 0.966... Calculate: c = 11.8 (to 1 decimal place)
Put in the values we know: a/sin A = 7/sin(35°) = c/sin(105°)
Ignore a/sin A (not useful to us): 7/sin(35°) = c/sin(105°)
Now we use our algebra skills to rearrange and solve:
Swap sides: c/sin(105°) = 7/sin(35°)
Multiply both sides by sin(105°): c = ( 7 / sin(35°) ) × sin(105°)
Calculate: c = ( 7 / 0.574... ) × 0.966... Calculate: c = 11.8 (to 1 decimal place)
Cosine
<------- FORMULA
Example:
Example:
We know angle C = 37º, a = 8 and b = 11
The Law of Cosines says: c2 = a2 + b2 - 2ab cos(C)
Put in the values we know: c2 = 82 + 112 - 2 × 8 × 11 × cos(37º)
Do some calculations: c2 = 64 + 121 - 176 × 0.798…
Which gives us: c2 = 44.44...
Take the square root: c = √44.44 = 6.67 (to 2 decimal places)
Answer: c = 6.67
The Law of Cosines says: c2 = a2 + b2 - 2ab cos(C)
Put in the values we know: c2 = 82 + 112 - 2 × 8 × 11 × cos(37º)
Do some calculations: c2 = 64 + 121 - 176 × 0.798…
Which gives us: c2 = 44.44...
Take the square root: c = √44.44 = 6.67 (to 2 decimal places)
Answer: c = 6.67
The Sum formula
1. sin(x + y) = sin x * cos y + cos x * sin y
2. cos(x + y) = cos x * cos y - sin x * sin y
3. tan(x + y) = [tan x + tan y] / [1 - tanx * tany]
2. cos(x + y) = cos x * cos y - sin x * sin y
3. tan(x + y) = [tan x + tan y] / [1 - tanx * tany]
The Difference formula
- sin(x - y) = sin x * cos y - cos x * sin y
- cos(x - y) = cos x * cos y + sin x * sin y
- tan(x - y) = [tan x - tan y] / [1 + tanx * tany]
The Double-Angle formula
sin(2A) = 2 sin A cos A
cos(2A) = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A
tan(2A) = 2 tan A / (1 − tan²A)
cos(2A) = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A
tan(2A) = 2 tan A / (1 − tan²A)
The Half Angle formula
sin(B/2) = ± sqrt([1 − cos B] / 2)
cos(B/2) = ± sqrt([1 + cos B] / 2)
tan(B/2) = (1 − cos B) / sin B = sin B / (1 + cos B)
cos(B/2) = ± sqrt([1 + cos B] / 2)
tan(B/2) = (1 − cos B) / sin B = sin B / (1 + cos B)