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Questions and Answers
What is the formula for the sine of a difference?
What is the formula for the sine of a difference?
- sin(α - β) = cos α cos β + sin α sin β
- sin(α - β) = sin α cos β - cos α sin β (correct)
- sin(α - β) = sin α cos β + cos α sin β
- sin(α - β) = cos α cos β - sin α sin β
What is the formula for the cosine of a sum?
What is the formula for the cosine of a sum?
- cos(α + β) = sin α cos β - cos α sin β
- cos(α + β) = sin α cos β + cos α sin β
- cos(α + β) = cos α cos β - sin α sin β (correct)
- cos(α + β) = cos α cos β + sin α sin β
What is the formula for the sine of a double angle?
What is the formula for the sine of a double angle?
- sin(2α) = sin α + cos α
- sin(2α) = sin α cos α
- sin(2α) = sin α - cos α
- sin(2α) = 2 sin α cos α (correct)
What is the formula for the cosine of a double angle?
What is the formula for the cosine of a double angle?
How can the formula for the sine of a difference be derived?
How can the formula for the sine of a difference be derived?
What is the correct expansion of the expression $\cos((90^\circ - \alpha) + eta)$ using the compound angle formula for cosine?
What is the correct expansion of the expression $\cos((90^\circ - \alpha) + eta)$ using the compound angle formula for cosine?
Which of the following is equivalent to the expression $\sin \alpha \cos eta + \cos \alpha \sin eta$?
Which of the following is equivalent to the expression $\sin \alpha \cos eta + \cos \alpha \sin eta$?
What is the value of $\sin(90^\circ - \alpha)$ in terms of a trigonometric function of $\alpha$?
What is the value of $\sin(90^\circ - \alpha)$ in terms of a trigonometric function of $\alpha$?
Which of the following is a correct expression for the cosine of a double angle?
Which of the following is a correct expression for the cosine of a double angle?
What is the value of $\sin(2\alpha)$ in terms of sine and cosine functions of $\alpha$?
What is the value of $\sin(2\alpha)$ in terms of sine and cosine functions of $\alpha$?
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Study Notes
Compound Angle Identities
- Cosine of a Difference:
\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta - Cosine of a Sum:
\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta - Sine of a Difference:
\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta - Sine of a Sum:
\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
Derivation of Compound Angle Formulas
- Derivation of
\cos(\alpha - \beta)uses the distance formula and cosine rule - Derivation of
\cos(\alpha + \beta)uses the negative angle identity and even-odd identities - Derivation of
\sin(\alpha - \beta)and\sin(\alpha + \beta)uses co-functions and compound angle formulas
Double Angle Formulas
- Sine of Double Angle:
\sin(2\alpha) = 2 \sin \alpha \cos \alpha - Cosine of Double Angle:
\cos(2\alpha) = \cos^2 \alpha - \sin^2 \alphaand alternative forms
Key Takeaways
- Compound angle identities relate to sums and differences of angles in trigonometry
- These identities are essential in solving triangular problems and other mathematical applications
Compound Angle Identities
- Cosine of a Difference:
\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta - Cosine of a Sum:
\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta - Sine of a Difference:
\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta - Sine of a Sum:
\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
Derivation of Compound Angle Formulas
- Derivation of
\cos(\alpha - \beta)uses the distance formula and cosine rule - Derivation of
\cos(\alpha + \beta)uses the negative angle identity and even-odd identities - Derivation of
\sin(\alpha - \beta)and\sin(\alpha + \beta)uses co-functions and compound angle formulas
Double Angle Formulas
- Sine of Double Angle:
\sin(2\alpha) = 2 \sin \alpha \cos \alpha - Cosine of Double Angle:
\cos(2\alpha) = \cos^2 \alpha - \sin^2 \alphaand alternative forms
Key Takeaways
- Compound angle identities relate to sums and differences of angles in trigonometry
- These identities are essential in solving triangular problems and other mathematical applications
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