Trigonometric Addition Formulas

Trigonometric addition formulas are a set of identities that show the relationship between the trigonometric functions of the sum or difference of two angles and the trigonometric functions of those angles themselves.

• Trigonometric Addition Formulas are an important tool in the study of trigonometry. 
• They allow us to express the trigonometric functions of the sum or difference of two angles in terms of the trigonometric functions of the angles themselves. 
• These formulas are derived from the Pythagorean identity and angle sum and difference identities 

Key Terms: Trigonometry, Angle, Sine, Cosine, Tangent, Addition, Subtraction, Theorem, Unit circle

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Trigonometric Addition Formulas

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The three most commonly used trigonometry addition formulas are the sine addition formula, cosine addition formula, and tangent addition formula. 

• The sine addition formula expresses the sine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. 
  • The formula is sin(a+b) = sin(a)cos(b) + cos(a)sin(b).

• The cosine addition formula expresses the cosine of the sum or difference of two angles in terms of the cosines and sines of the individual angles. 
  • The formula is cos(a+b) = cos(a)cos(b) - sin(a)sin(b). 
• The tangent addition formula expresses the tangent of the sum or difference of two angles in terms of the tangents of the individual angles. 
  • The formula is tan(a+b) = (tan(a) + tan(b))/(1 - tan(a)tan(b)).

Formulas

There are three main trigonometric addition formulas: the sine addition formula, cosine addition formula, and tangent addition formula. 

Sine Addition Formula: 

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Sine Subtraction Formula:

sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Cosine Addition Formula:

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

Cosine Subtraction Formula:

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

Tangent Addition Formula:

tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

Tangent Subtraction Formula:

tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b))

In addition to these six main formulas, there are also three double angle formulas:

• sin(2a) = 2sin(a)cos(a)
• cos(2a) = cos²(a) - sin²(a) = 2cos²(a) - 1 = 1 - 2sin²(a)
• tan(2a) = (2tan(a)) / (1 - tan²(a))

These formulas relate the trigonometric functions of double angles to the trigonometric functions of the original angles.

Furthermore, there are also three half-angle formulas:

• sin(a/2) = ±√[(1 - cos(a)) / 2]
• cos(a/2) = ±√[(1 + cos(a)) / 2]
• tan(a/2) = ±√[(1 - cos(a)) / (1 + cos(a))]

These formulas relate the trigonometric functions of half angles to the trigonometric functions of the original angles.

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Solved Examples

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Example 1: Use the sine addition formula to find sin(75°).

Solution: The sine addition formula is used as follows:

sin(75°) = sin(45° + 30°) = sin(45°) cos(30°) + cos(45°) sin(30°)

Substituting sin(45°) = √2/2 and cos(30°) = √3/2 values in the above equation,

sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2)

Simplifying:

sin(75°) = (√6 + √2) / 4

Therefore, sin(75°) = (√6 + √2) / 4.

Example 2: Use the cosine addition formula to find cos(75°).

Solution: The cosine addition formula is used to solve this question as follows:

cos(75°) = cos(45° + 30°) = cos(45°) cos(30°) - sin(45°) sin(30°)

Substituting cos(45°) = √2/2 and sin(30°) = 1/2 values in the above equation,

cos(75°) = (√2/2)(√3/2) - (√2/2)(1/2)

Simplifying:

cos(75°) = (√6 - √2) / 4

Therefore, cos(75°) = (√6 - √2) / 4.

Example 3: Use the tangent addition formula to find the value of tan(75°).

Solution: The tangent addition formula is:

tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

We can use this formula to find the value of tan(75°) as follows:

tan(75°) = tan(45° + 30°) = (tan(45°) + tan(30°)) / (1 - tan(45°)tan(30°))

Substituting tan(30°) = 1/√3 and tan(45°) = 1 values in the above equation,

tan(75°) = (1 + 1/√3) / (1 - 1/√3)

Simplifying:

tan(75°) = (3 + √3) / (3 - √3)

Rationalizing the denominator by multiplying both numerator and denominator by the conjugate of the denominator:

tan(75°) = [(3 + √3) / (3 - √3)] * [(3 + √3) / (3 + √3)]

tan(75°) = (9 + 6√3 + 3) / (9 - 3)

tan(75°) = (12 + 6√3)

Therefore, tan(75°) = 12 + 6√3.

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Value of Sin 120 with verification

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Using the sine addition formula, we can find the value of sin(120°) as follows:

sin(120°) = sin(90° + 30°) = sin(90°)cos(30°) + cos(90°)sin(30°)

Since sin(90°) = 1 and cos(90°) = 0, we have:

sin(120°) = cos(30°)

Using the exact value of cos(30°) = √3/2, we get:

sin(120°) = √3/2

Therefore, sin(120°) = √3/2.

Verification: To verify the value of sin(120°) using the quadrant relationship, note that in the second quadrant, the sine function is positive and the cosine function is negative. Therefore:

sin(120°) = |sin(120°)| = sin(60°)

Using the exact value of sin(60°) = √3/2, we have:

sin(120°) = √3/2

This is consistent with the value obtained earlier using the sine addition formula.

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Derivation of Trigonometric Addition Formulas

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There are several ways to derive these formulas, but one common method is to use the complex exponential function.

Let's start by defining the complex exponential function:

e^(ix) = cos(x) + i sin(x)

where i is the imaginary unit (i² = -1).

Using this definition, we can derive the following addition 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 + b) = (tan(a) + tan(b)) / (1 - tan(a) tan(b))

To derive the sine formula, we start with:

e^(i(a+b)) = e^(ia) e^(ib)

Expanding both sides using Euler's formula (which relates the exponential function to sine and cosine), we get:

cos(a+b) + i sin(a+b) = (cos(a) + i sin(a))(cos(b) + i sin(b))

Expanding the right-hand side using the distributive property, we get:

cos(a+b) + i sin(a+b) = cos(a) cos(b) + i sin(a) cos(b) + i sin(b) cos(a) - sin(a) sin(b)

Equating the real and imaginary parts of both sides, we get:

sin(a+b) = sin(a) cos(b) + cos(a) sin(b)

cos(a+b) = cos(a) cos(b) - sin(a) sin(b)

To derive the cosine formula, we use the same approach but use the imaginary part instead of the real part. This gives:

cos(a+b) + i sin(a+b) = (cos(a) + i sin(a))(cos(b) + i sin(b))

cos(a+b) + i sin(a+b) = cos(a) cos(b) - sin(a) sin(b) + i sin(a) cos(b) + i cos(a) sin(b)

Equating the real and imaginary parts of both sides, we get:

cos(a+b) = cos(a) cos(b) - sin(a) sin(b)

sin(a+b) = sin(a) cos(b) + cos(a) sin(b)

To derive the tangent formula, we divide the sine formula by the cosine formula:

tan(a+b) = (sin(a+b))/(cos(a+b))

tan(a+b) = (sin(a) cos(b) + cos(a) sin(b))/(cos(a) cos(b) - sin(a) sin(b))

Using the identities sin(a+b) and cos(a+b), we get:

tan(a+b) = (tan(a) + tan(b))/(1 - tan(a) tan(b))

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Proof of Sin(a+b) Theorem

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Consider two angles, a and b, such that 0° ≤ a, b ≤ 90°. 

Let A be the point on the unit circle that corresponds to angle a, and let B be the point on the unit circle that corresponds to angle b. Let C be the foot of the perpendicular from A to the x-axis, and let D be the foot of the perpendicular from B to the x-axis. Let E be the intersection point of the line segments AC and BD.

Therefore, AC = sin(a) and BD = sin(b). And AE = cos(b) and BE = cos(a), since these are the x-coordinates of points A and B, respectively. Furthermore, CE = sin(b)cos(a) and DE = sin(a)cos(b), since these are the y-coordinates of points C and D, respectively.

Now calculate the length of line segment AE using the Pythagorean theorem:

AE² = AC² + CE²

AE² = sin²(a) + sin²(b)cos²(a)

Similarly, calculate the length of line segment BE:

BE² = BD² + DE²

BE² = sin²(b) + sin²(a)cos²(b)

Now, calculate the length of line segment AB using the Pythagorean theorem:

AB² = AE² + BE²

AB² = sin²(a) + sin²(b)cos²(a) + sin²(b) + sin²(a)cos²(b)

Simplifying and factoring out sin²(a)sin²(b), we get:

AB² = sin²(a)(1 - cos²(b)) + sin²(b)(1 - cos²(a))

AB² = sin²(a)sin²(b) + (sin²(a)cos^2(b) + sin²(b)cos²(a))

AB² = sin²(a)sin²(b) + 2sin(a)sin(b)cos(a)cos(b)

Taking the square root of both sides, we get:

AB = sin(a)cos(b) + cos(a)sin(b)

Therefore, we have proved the sine addition formula:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

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Things to Remember

1. Trigonometric addition formulas are used to find the sine, cosine, and tangent of the sum or difference of two angles.
2. The two most common trigonometric addition formulas are the sum and difference formulas, which are:
   • sin(A + B) = sin A cos B + cos A sin B
   • cos(A + B) = cos A cos B - sin A sin B
3. The sum and difference formulas can be used in reverse to find the sine, cosine, and tangent of the difference of two angles.
4. To use the addition formulas, it is important to be familiar with the values of sine and cosine for common angles, such as 0, 30, 45, 60, and 90 degrees.
5. Trigonometric addition formulas are often used in calculus, physics, and engineering to solve problems related to angles and periodic functions.