10 Questions
What is the formula for the sine of a difference?
sin(α - β) = sin α cos β - cos α sin β
What is the formula for the cosine of a sum?
cos(α + β) = cos α cos β - sin α sin β
What is the formula for the sine of a double angle?
sin(2α) = 2 sin α cos α
What is the formula for the cosine of a double angle?
cos(2α) = cos² α - sin² α
How can the formula for the sine of a difference be derived?
Using the compound angle formula for cosine and co-functions
What is the correct expansion of the expression $\cos((90^\circ - \alpha) + eta)$ using the compound angle formula for cosine?
$\cos(90^\circ - \alpha) \cos eta - \sin(90^\circ - \alpha) \sin eta$
Which of the following is equivalent to the expression $\sin \alpha \cos eta + \cos \alpha \sin eta$?
$\sin(\alpha + eta)$
What is the value of $\sin(90^\circ - \alpha)$ in terms of a trigonometric function of $\alpha$?
$\cos \alpha$
Which of the following is a correct expression for the cosine of a double angle?
$\cos^2 \alpha - \sin^2 \alpha$
What is the value of $\sin(2\alpha)$ in terms of sine and cosine functions of $\alpha$?
$2\sin \alpha \cos \alpha$
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
Test your knowledge of compound angle identities in trigonometry, including formulas for cosine and sine of a difference and sum.
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