Trigonometry Formulas and Identities
Trigonometry Formula is a set of mathematical equations and relationships that are used to solve problems related to angles and sides of triangles. Trigonometry formulas are fundamental tools in the field of trigonometry and are used extensively in various disciplines such as mathematics, physics, engineering, and navigation. All Trigonometry formulas help in calculating the values of unknown angles or sides based on the given information.
The following is a list of all Trigonometry Formulas.
Tangent and Cotangent Formulas
sin θ
tan θ = ———————
cos θ
cos θ
cot θ = ———————
sin θ
Even and Odd functions
sin(-θ) = - sinθ cos(-θ) = cosθ tan(-θ) = - tanθ cosec(-θ) = - cosecθ sec(-θ) = secθ cot(-θ) = - cotθ
Reciprocal Identities
1
sin θ = —————————
cosec θ
1
cos θ = —————————
sec θ
1
tan θ = —————————
cot θ
1
cosec θ = ———————
sin θ
1
sec θ = ———————
cos θ
1
cot θ = ———————
tan θ
Pythagorean Identities
sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ
Periodic Formulas
If ‘n’ is an integer, then sin(θ+2πn) = sinθ cos(θ+2πn) = cosθ tan(θ+2πn) = tanθ cosec(θ+2πn) = cosecθ sec(θ+2πn) = secθ cot(θ+2πn) = cotθ
Sum and Difference Formulas
sin(A±B) = sin A cos B ± cos A sin B
cos(A±B) = cos A cos B ± sin A sin B
tan A ± tan B
tan(A±B) = —————————————————
1 ± tan A tan B
Product to sum Formulas
1
sin A sin B = — [cos(A-B)-cos(A+B)]
2
1
cos A cos B = — [cos(A-B)+cos(A+B)]
2
1
sin A cos B = — [sin(A+B)+sin(A-B)]
2
1
cos A sin B = — [sin(A+B)-sin(A-B)]
2
Co Function Formulas
π
sin(——— - θ) = cos θ
2
π
cos(——— - θ) = sin θ
2
π
tan(——— - θ) = cot θ
2
π
sec(——— - θ) = cosec θ
2
π
cosec(——— - θ) = sec θ
2
π
cot(——— - θ) = tan θ
2
Multiple and Half angles
1. sin2A = 2sinAcosA
2. cos2A = cos²A - sin²A
= cos²A - 1
= 1 - 2sin²A
2tanA
3. tan2A = ———————————
1 - tan²A
2tanA
4. sin2A = ———————————
1 + tan²A
1 - tan²A
5. cos2A = ———————————
1 + tan²A
6. sin3A = 3sinA - 4sin³A
7. cos3A = 3cos³A - 3cosA
3tanA - tan³A
8. tan3A = ———————————————
1 - 3tan²A
Factorization of the Sum or Difference of sine and cosine with two variables
C + D C - D
1. sinC + sinD = 2sin(———————)cos(———————)
2 2
C + D C - D
2. sinC - sinD = 2cos(———————)sin(———————)
2 2
C + D C - D
3. cosC + cosD = 2cos(———————)cos(———————)
2 2
C + D C - D
4. cosC - cosD = -2sin(———————)sin(———————)
2 2
Trigonometry Table
|
Angle(Degrees) |
0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° |
| Angle(Radians) | 0 | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | 5π/6 | π |
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1/2 | -1/√2 | -√3/2 | -1 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ | -√3 | -1 | -1/√3 | 0 |
| cot | ∞ | √3 | 1 | 1/√3 | 0 | -1/√3 | -1 | -√3 | ∞ |
| sec | 1 | 2/√3 | √2 | 2 | ∞ | -2 | -√2 | -2/√3 | -1 |
| cosec | ∞ | 2 | √2 | 2/√3 | 1 | 2/√3 | √2 | 2 | ∞ |
Trigonometry Formula Topics
Trigonometry Formula and Identities
Trigonometry Tangent and Cotangent Formulas
Trigonometry Even and Odd functions
Trigonometry Reciprocal Identities
Trigonometry Pythagorean Identities
Trigonometry Pythagorean Even and Odd functions
Trigonometry Periodic Formulas
Trigonometry Double Angle Formulas
Trigonometry Sum and Difference Formulas
Trigonometry Product to sum Formulas
Trigonometry Co Function Formulas
Trigonometry Problems
- If tan(theta)=1/√7, Evaluate (cosec^2 (theta)-sec^2 (theta)/ (cosec^2 (theta)+sec^2 (theta))
- If 3 tan(theta) = 4, find the value of (5 sin(theta)-3cos(theta)) / (5 sin(theta)+2cos(theta))
- If sec(theta)=sqrt(2), Evaluate (1+tan(theta)+cosec(theta))/(1+cot(theta)-cosec(theta))
- If 3 cot A = 4, Check whether (1-tan^2A)/(1+tan^2A) = cos^2 A – sin^2 A or not
- Prove that tan A/ (1-cot A) +cot A / (1-tan A) = sec A cosec A+1
- If tan(theta) + cot(theta)=5, find the value of tan^2 (theta)+cot^2 (theta)
- Prove that sqrt((1-sin(theta))/(1+sin(theta))) = sec(theta) – tan(theta)
- If sec A + tan A=7, find the value of sec A – tan A
- Given that tan (theta) = p/q. Find the value of (p sin(theta)-q cos (theta)) / (p sin (theta)+q cos (theta))
- If tan A=1/2 and tan B=1/3 then find the value of sin(A+B) using the formula: tan(A+B)=(tan A + tan B)/(1-tan A tan B)