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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Sign up for freeWhat 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?
What is the tangent rule used for?
How does one correctly apply the tangent rule to solve a problem?
What is the tangent rule formula for the sum of two angles?
What steps should be followed to use the tangent rule effectively?
What is the tangent of 75° using the tangent-sum formula for 30° and 45°?
How should you start using the tangent rule in a problem?
What is the tangent of the sum of 30° and 45° using the tangent rule?
What is the tangent rule for the sum of two angles?
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?
What is the tangent rule used for?
How does one correctly apply the tangent rule to solve a problem?
What is the tangent rule formula for the sum of two angles?
What steps should be followed to use the tangent rule effectively?
What is the tangent of 75° using the tangent-sum formula for 30° and 45°?
How should you start using the tangent rule in a problem?
What is the tangent of the sum of 30° and 45° using the tangent rule?
What is the tangent rule for the sum of two angles?
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Understanding the tangent rule is an important step in enhancing your mathematical skills. It is particularly useful in the field of trigonometry and calculus.
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, \[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
To effectively use the tangent rule, you need to follow a series of steps:
Remember that the tangent rule only applies to angles measured in radians or degrees. Ensure you are consistent with your angle measurements.
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:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
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.
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, \[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
To effectively use the tangent rule, follow these steps:
Remember that the tangent rule only applies to angles measured in radians or degrees. Ensure you are consistent with your angle measurements.
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:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
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.
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:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
To effectively use the tangent rule, follow these steps:
Remember that the tangent rule only applies to angles measured in radians or degrees. Ensure you are consistent with your angle measurements.
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:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
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.
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:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
To effectively use the tangent rule, follow these steps:
Ensure that your angle measurements are consistent. The tangent rule applies to both radians and degrees.
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:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
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