# Tangent rule

The tangent rule, also known as the tangent-secant theorem, states that the tangent to a circle is perpendicular to the radius at the point of contact. Additionally, it asserts that the product of the lengths of the entire secant segment and its external segment is equal for any two secants intersecting outside a circle. This geometric principle is instrumental in solving various problems involving circles, tangents, and secants.

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## Tangent Rule in Mathematics

Understanding the tangent rule is an important step in enhancing your mathematical skills. It is particularly useful in the field of trigonometry and calculus.

### Definition of the Tangent Rule

The tangent rule, also known as the tangent-sum formula, is a crucial trigonometric identity used to find the tangent of the sum or difference of two angles. It is expressed as:

For the sum of two angles, $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

For the difference of two angles, $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

### How to Use the Tangent Rule

To effectively use the tangent rule, you need to follow a series of steps:

• Identify the angles involved in the problem.
• Determine whether you are dealing with the sum or difference of these angles.
• Apply the appropriate tangent formula.
• Simplify the expression to arrive at the solution.

Remember that the tangent rule only applies to angles measured in radians or degrees. Ensure you are consistent with your angle measurements.

### Examples of the Tangent Rule

Example 1: Find the tangent of the sum of 30° and 45°.

Using the tangent rule:

$\tan(30° + 45°) = \frac{\tan 30° + \tan 45°}{1 - \tan 30° \tan 45°}$

We know that $$\tan 30° = \frac{1}{\sqrt{3}}$$ and $$\tan 45° = 1$$, so:

$\tan(30° + 45°) = \frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}} \cdot 1}$

Simplify this to get:

$\tan 75° = \frac{\frac{1 + \sqrt{3}}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}}$

Which simplifies further to:

$\tan 75° = \frac{1 + \sqrt{3}}{\sqrt{3} - 1}$

Understanding the derivation of the tangent rule can provide a deeper appreciation for this formula. The tangent rule is derived using the sine and cosine addition formulas. For the sum:

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

Dividing these two equations gives:

$\frac{\sin(A + B)}{\cos(A + B)} = \frac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B}$

Since $$\frac{\sin(A + B)}{\cos(A + B)} = \tan(A + B)$$, we arrive at the tangent-sum formula:

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

## Definition of Tangent Rule

The tangent rule is an essential concept in trigonometry that helps in finding the tangent of the sum or difference of two angles. Whether you are solving complex equations or simple problems, this rule is highly valuable.

### Definition of the Tangent Rule

The tangent rule, also known as the tangent-sum formula, is expressed as:For the sum of two angles, $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ For the difference of two angles, $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

### How to Use the Tangent Rule

To effectively use the tangent rule, follow these steps:

• Identify the angles involved in the problem.
• Determine whether you are dealing with the sum or difference of these angles.
• Apply the appropriate tangent formula.
• Simplify the expression to arrive at the solution.

Remember that the tangent rule only applies to angles measured in radians or degrees. Ensure you are consistent with your angle measurements.

### Examples of the Tangent Rule

Example 1: Find the tangent of the sum of 30° and 45°.

Using the tangent rule:

$\tan(30° + 45°) = \frac{\tan 30° + \tan 45°}{1 - \tan 30° \tan 45°}$

We know that $$\tan 30° = \frac{1}{\sqrt{3}}$$ and $$\tan 45° = 1$$, so:

$\tan(30° + 45°) = \frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}} \cdot 1}$

Simplify this to get:

$\tan 75° = \frac{\frac{1 + \sqrt{3}}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}}$

Which simplifies further to:

$\tan 75° = \frac{1 + \sqrt{3}}{\sqrt{3} - 1}$

Understanding the derivation of the tangent rule can provide a deeper appreciation for this formula. The tangent rule is derived using the sine and cosine addition formulas.For the sum:

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

Dividing these two equations gives:

$\frac{\sin(A + B)}{\cos(A + B)} = \frac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B}$

Since $$\frac{\sin(A + B)}{\cos(A + B)} = \tan(A + B)$$, we arrive at the tangent-sum formula:

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

## Tangent Rule Explained with Examples

Understanding the tangent rule is an integral part of enhancing your mathematical skills. This trigonometric identity is especially useful in solving problems related to the sum or difference of angles.

### Definition of the Tangent Rule

The tangent rule, also known as the tangent-sum formula, is expressed as:

For the sum of two angles:

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

For the difference of two angles:

$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

### How to Use the Tangent Rule

To effectively use the tangent rule, follow these steps:

• Identify the angles involved in the problem.
• Determine whether you are dealing with the sum or difference of these angles.
• Apply the appropriate tangent formula.
• Simplify the expression to arrive at the solution.

Remember that the tangent rule only applies to angles measured in radians or degrees. Ensure you are consistent with your angle measurements.

### Examples of the Tangent Rule

Example 1: Find the tangent of the sum of 30° and 45°.

Using the tangent rule:

$\tan(30° + 45°) = \frac{\tan 30° + \tan 45°}{1 - \tan 30° \tan 45°}$

We know that $$\tan 30° = \frac{1}{\sqrt{3}}$$ and $$\tan 45° = 1$$, so:

$\tan(30° + 45°) = \frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}} \cdot 1}$

Simplify this to get:

$\tan 75° = \frac{\frac{1 + \sqrt{3}}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}}$

Which further simplifies to:

$\tan 75° = \frac{1 + \sqrt{3}}{\sqrt{3} - 1}$

Understanding the derivation of the tangent rule can provide a deeper appreciation for this formula. The tangent rule is derived using the sine and cosine addition formulas.

For the sum:

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

Dividing these two equations gives:

$\frac{\sin(A + B)}{\cos(A + B)} = \frac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B}$

Since $$\frac{\sin(A + B)}{\cos(A + B)} = \tan(A + B)$$, we arrive at the tangent-sum formula:

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

## Tangent Rule in Trigonometry Techniques

The tangent rule is a powerful tool in trigonometry that allows you to calculate the tangent of the sum or difference of angles. Knowing how to apply this rule can simplify many trigonometric problems. Let's explore the tangent rule in detail.

### Definition of the Tangent Rule

The \textbf{tangent rule}, or tangent-sum formula, is given by:

For the sum of two angles:

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

For the difference of two angles:

$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

### How to Use the Tangent Rule

To effectively use the tangent rule, follow these steps:

• Identify the angles involved in the problem.
• Determine whether you are dealing with the sum or difference of these angles.
• Apply the appropriate tangent formula.
• Simplify the expression to arrive at the solution.

Ensure that your angle measurements are consistent. The tangent rule applies to both radians and degrees.

### Examples of the Tangent Rule

Example 1: Find the tangent of the sum of 30° and 45°.

Using the tangent rule:

$\tan(30° + 45°) = \frac{\tan 30° + \tan 45°}{1 - \tan 30° \tan 45°}$

We know that $$\tan 30° = \frac{1}{\sqrt{3}}$$ and $$\tan 45° = 1$$, so:

$\tan(30° + 45°) = \frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}} \cdot 1}$

Simplify this to get:

$\tan 75° = \frac{\frac{1 + \sqrt{3}}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}}$

Which further simplifies to:

$\tan 75° = \frac{1 + \sqrt{3}}{\sqrt{3} - 1}$

Understanding the derivation of the tangent rule can provide a deeper appreciation for this formula. The tangent rule is derived using the sine and cosine addition formulas.

For the sum:

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

Dividing these two equations gives:

$\frac{\sin(A + B)}{\cos(A + B)} = \frac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B}$

Since $$\frac{\sin(A + B)}{\cos(A + B)} = \tan(A + B)$$, we arrive at the tangent-sum formula:

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

## Tangent rule - Key takeaways

• Tangent Rule Definition: The tangent rule, also known as the tangent-sum formula, is a trigonometric identity used to find the tangent of the sum or difference of two angles.
• Sum Formula: For the sum of two angles, the tangent rule is expressed as: $$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$.
• Difference Formula: For the difference of two angles, the tangent rule is given by: $$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$.
• Usage Steps: Identify the angles, determine whether dealing with sum or difference, apply the appropriate tangent formula, and simplify the expression.
• Example Application: Finding $$\tan(30° + 45°)$$ using the tangent rule results in $$\tan 75° = \frac{1 + \sqrt{3}}{\sqrt{3} - 1}$$.

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What is the tangent rule in trigonometry?
The tangent rule in trigonometry relates the tangent of angles in a triangle to the sides of the triangle. It states that for any triangle, the ratio of the length of a side to the tangent of the opposite angle is constant: \$$\\frac{a}{\\tan A} = \\frac{b}{\\tan B} = \\frac{c}{\\tan C} \$$.
How is the tangent rule applied to solve triangles?
The tangent rule, also known as the tangent formula, is used in trigonometry to find unknown sides or angles in non-right-angled triangles. It states that (a-b)/(a+b) = tan[(A-B)/2]/tan[(A+B)/2], where a and b are sides opposite angles A and B, respectively. This helps solve triangles by relating the sides and angles.
What is an intuitive explanation of the tangent rule?
The tangent rule relates the angles and sides in a non-right-angled triangle, stating that the ratio of the tangent of half the sum of two angles to the tangent of half their difference is equal to the ratio of the lengths of the opposite sides. This helps in solving triangles when certain angle relationships are known.
What are some real-world applications of the tangent rule?
The tangent rule is applied in surveying for determining distances and angles, in navigation for plotting courses on maps, in architecture for designing angles and slopes, and in physics for analysing forces and motion in mechanics.
How does the tangent rule relate to other trigonometric rules?
The tangent rule relates to other trigonometric rules by expressing the tangent of an angle in a triangle in terms of the sides, akin to the sine and cosine rules. It complements the sine rule and cosine rule by solving for different angles or sides using tangent relationships.

## Test your knowledge with multiple choice flashcards

What is the formula for the tangent of the sum of two angles?

What is the first step in using the tangent rule?

What is the general formula for the tangent of the sum of two angles?

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