The half angle identities are a set of trigonometric identities that are used to find the values of trigonometric functions of half angles. These identities are useful in a variety of applications, including calculus, physics, and engineering. In this article, we will explore the five half angle identities for sine, cosine, and tangent, and provide examples of how to use them to solve problems.
Key Points
- The half angle identities for sine, cosine, and tangent can be used to find the values of these functions for half angles.
- These identities are useful in a variety of applications, including calculus, physics, and engineering.
- The half angle identities can be derived using the double angle identities and the Pythagorean identity.
- There are five half angle identities, one for each of the following: sine, cosine, tangent, cotangent, and secant.
- These identities can be used to simplify trigonometric expressions and to solve trigonometric equations.
Half Angle Identity for Sine
The half angle identity for sine is given by the formula: sin(α/2) = ±√[(1 - cos(α))/2]. This identity can be used to find the value of the sine function for half angles. For example, to find the value of sin(30°/2), we can use the half angle identity for sine: sin(15°) = ±√[(1 - cos(30°))/2] = ±√[(1 - √3/2)/2].
Derivation of the Half Angle Identity for Sine
The half angle identity for sine can be derived using the double angle identity for sine and the Pythagorean identity. The double angle identity for sine is given by the formula: sin(2α) = 2sin(α)cos(α). Using this identity, we can derive the half angle identity for sine: sin(α/2) = ±√[(1 - cos(α))/2].
| Angle | Sine Value |
|---|---|
| 30° | 1/2 |
| 15° | ±√[(1 - √3/2)/2] |
Half Angle Identity for Cosine
The half angle identity for cosine is given by the formula: cos(α/2) = ±√[(1 + cos(α))/2]. This identity can be used to find the value of the cosine function for half angles. For example, to find the value of cos(60°/2), we can use the half angle identity for cosine: cos(30°) = ±√[(1 + cos(60°))/2] = ±√[(1 + 1⁄2)/2] = ±√(3⁄4).
Derivation of the Half Angle Identity for Cosine
The half angle identity for cosine can be derived using the double angle identity for cosine and the Pythagorean identity. The double angle identity for cosine is given by the formula: cos(2α) = 2cos^2(α) - 1. Using this identity, we can derive the half angle identity for cosine: cos(α/2) = ±√[(1 + cos(α))/2].
| Angle | Cosine Value |
|---|---|
| 60° | 1/2 |
| 30° | ±√(3/4) |
Half Angle Identity for Tangent
The half angle identity for tangent is given by the formula: tan(α/2) = sin(α)/(1 + cos(α)) or tan(α/2) = (1 - cos(α))/sin(α). This identity can be used to find the value of the tangent function for half angles. For example, to find the value of tan(45°/2), we can use the half angle identity for tangent: tan(22.5°) = sin(45°)/(1 + cos(45°)) = (1/√2)/(1 + 1/√2).
Derivation of the Half Angle Identity for Tangent
The half angle identity for tangent can be derived using the double angle identity for tangent and the Pythagorean identity. The double angle identity for tangent is given by the formula: tan(2α) = 2tan(α)/(1 - tan^2(α)). Using this identity, we can derive the half angle identity for tangent: tan(α/2) = sin(α)/(1 + cos(α)) or tan(α/2) = (1 - cos(α))/sin(α).
| Angle | Tangent Value |
|---|---|
| 45° | 1 |
| 22.5° | (1/√2)/(1 + 1/√2) |
Half Angle Identity for Cotangent
The half angle identity for cotangent is given by the formula: cot(α/2) = (1 + cos(α))/sin(α) or cot(α/2) = sin(α)/(1 - cos(α)). This identity can be used to find the value of the cotangent function for half angles. For example, to find the value of cot(30°/2), we can use the half angle identity for cotangent: cot(15°) = (1 + cos(30°))/sin(30°) = (1 + √3/2)/(1⁄2).
Derivation of the Half Angle Identity for Cotangent
The half angle identity for cotangent can be derived using the double angle identity for cotangent and the Pythagorean identity. The double angle identity for cotangent is given by the formula: cot(2α) = (cot^2(α) - 1)/(2cot(α)). Using this identity, we can derive the half angle identity for cotangent: cot(α/2) = (1 + cos(α))/sin(α) or cot(α/2) = sin(α)/(1 - cos(α)).
| Angle | Cotangent Value |
|---|---|
| 30° | √3 |
| 15° | (1 + √3/2)/(1/2) |
Half Angle Identity for Secant
The half angle identity for secant is given by the formula: sec(α/2) = (1 + cos(α))/cos(α) or sec(α/2) = 1/√[(1 + cos(α))/2]. This identity can be used to find the value of the secant function for half angles. For example, to find the value of sec(60°/2), we can use the half angle identity for secant: sec(30°) = (1 + cos(60°))/cos(60°) = (1 + 1⁄2)/(1⁄2) = 2⁄1.
Derivation of the Half Angle Identity for Secant
The half angle identity for secant can be derived using the double angle identity for secant and the Pythagorean identity. The double angle identity for secant is given by the formula: sec(2α) = sec^2(α)/(2 - sec^2(α)). Using this identity, we can derive the half angle identity for secant: sec(α/2) = (1 + cos(α))/cos(α) or sec(α/2) = 1/√[(1 + cos(α))/2].
| Angle | Secant Value |
|---|---|
| 60° | 2 |
| 30° | 2/1 |
What are the five half angle identities?
+The five half angle identities are: sin(α/2) = ±√[(1 - cos(α))/2], cos(α/2) = ±√[(1 + cos(α))/2], tan(α/2) = sin(α)/(1 + cos(α)) or tan(α/2) = (1 - cos(α))/sin(α), cot(α/2) = (1 + cos(α))/sin(α) or cot(α/2) = sin(α)/(1 - cos(α)), and sec(α/2) = (1 + cos(α))/cos(α) or sec(α/2) = 1/√[(1 + cos(α))/2].
How are the half angle identities derived?
+The half angle identities can be derived using the double angle identities and the Pythagorean identity. For example, the half angle identity for sine can be derived using the double angle identity for sine and the Pythagorean identity.
What are some common applications of the half angle identities?
+The half angle identities have a variety of applications, including calculus, physics, and engineering. For example, they can be used to find the value of trigonometric functions for half angles in problems involving right triangles.
