Question

7 THE EXPONENTIAL FORMS OF SIN AND COS: Euler's formula can be used to write these functions in exponential form. They are: $\cos \theta = \frac{1}{2} (e^{i\theta} + e^{-i\theta})$ and $\sin \theta = \frac{1}{2i} (e^{i\theta} - e^{-i\theta})$ 8. (a) Derive the exponential forms of $\cos \theta$ and $\sin \theta$ given in Box 7. (b) Use these results to verify the following trigonometric identities. (i) $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$ (ii) $\sin 2\theta = 2 \cos \theta \sin \theta$ (iii) $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ (iv) $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

          7
THE EXPONENTIAL FORMS OF SIN AND COS: Euler's formula can be used to write these
functions in exponential form. They are:
$\cos \theta = \frac{1}{2} (e^{i\theta} + e^{-i\theta})$ and $\sin \theta = \frac{1}{2i} (e^{i\theta} - e^{-i\theta})$
8. (a) Derive the exponential forms of $\cos \theta$ and $\sin \theta$ given in Box 7.
(b) Use these results to verify the following trigonometric identities.
(i) $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
(ii) $\sin 2\theta = 2 \cos \theta \sin \theta$
(iii) $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$
(iv) $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

7
THE EXPONENTIAL FORMS OF SIN AND COS: Euler's formula can be used to write these
functions in exponential form. They are:
cosθ = (1)/(2) (e^iθ + e^-iθ) and sinθ = (1)/(2i) (e^iθ - e^-iθ)
8. (a) Derive the exponential forms of cosθ and sinθ given in Box 7.
(b) Use these results to verify the following trigonometric identities.
(i) cos 2θ = cos^2 θ - sin^2 θ
(ii) sin 2θ = 2 cosθsinθ
(iii) cos(α + β) = cosαcosβ - sinαsinβ
(iv) sin(α + β) = sinαcosβ + cosαsinβ

Added by Melissa L.

Question

Please give Ace some feedback