Textbook Question
Prove the following identities.
0. Functions
Trigonometric Identities
7:38 minutesProblem 56
Textbook Question
Derive a formula for tan (A − B).
Verified step by step guidance1
Start by recalling the identity for tangent of a difference: \( \tan(A - B) = \frac{\sin(A - B)}{\cos(A - B)} \).
Use the angle subtraction identities for sine and cosine: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \) and \( \cos(A - B) = \cos A \cos B + \sin A \sin B \).
Substitute these identities into the formula for \( \tan(A - B) \): \( \tan(A - B) = \frac{\sin A \cos B - \cos A \sin B}{\cos A \cos B + \sin A \sin B} \).
To simplify, divide both the numerator and the denominator by \( \cos A \cos B \): \( \tan(A - B) = \frac{\frac{\sin A}{\cos A} - \frac{\sin B}{\cos B}}{1 + \frac{\sin A}{\cos A} \cdot \frac{\sin B}{\cos B}} \).
Recognize that \( \frac{\sin A}{\cos A} = \tan A \) and \( \frac{\sin B}{\cos B} = \tan B \), leading to the final formula: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
7mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving equations in trigonometry. Key identities include the Pythagorean identities, reciprocal identities, and angle sum/difference identities, which are crucial for deriving formulas like tan(A - B).
Recommended video:
7:17
Verifying Trig Equations as Identities
Angle Difference Formula
The angle difference formula for tangent states that tan(A - B) can be expressed in terms of the tangents of angles A and B. Specifically, it is given by the formula tan(A - B) = (tan A - tan B) / (1 + tan A * tan B). This formula is derived from the sine and cosine definitions of tangent and is fundamental for solving problems involving the difference of angles.
Recommended video:
Guided course
5:04Converting between Degrees & Radians
Derivation Techniques
Derivation techniques in calculus involve using algebraic manipulation and known identities to derive new formulas or results. In the context of trigonometric functions, this often includes substituting known identities and simplifying expressions. Mastery of these techniques is essential for effectively deriving formulas like tan(A - B) and understanding their applications in various mathematical contexts.
Recommended video:
05:44Derivatives
6:36mWatch next
Master Simplifying Trig Expressions with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice