Consider two unit vectors a and b, as shown in Fig. 1-19....

Consider two unit vectors a and b, as shown in Fig. 1-19. Find the trigonometric relations for the sine and cosine of the sum and diference of two angles from the values of $\mathbf{a} \cdot \mathbf{b}$ and $\mathbf{a} \times \mathbf{b}$. Solution: First, $$ \begin{aligned} &\mathbf{a}=\cos \alpha \mathbf{i}+\sin \alpha \mathbf{j} \\ &\mathbf{b}=\cos \beta \mathbf{i}+\sin \beta \mathbf{j} \end{aligned} $$ Now we can write the product $\mathbf{a} \cdot \mathbf{b}$ in two different ways. From Eq. 1-11, $$ \mathbf{a} \cdot \mathbf{b}=\cos \alpha \cos \beta+\sin \alpha \sin \beta $$ and, from Eq. 1-5, $$ \mathbf{a} \cdot \mathbf{b}=\cos (\alpha-\beta) $$ Thus $$ \cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta $$ If we set $\beta^{\prime}=-\beta$ $$ \cos \left(\alpha+\beta^{\prime}\right)=\cos \alpha \cos \beta^{\prime}-\sin \alpha \sin \beta^{\prime} $$ Similarly, from Eq. 1-22, $$ \begin{aligned} \mathbf{a} \times \mathbf{b}=& \mid \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \cos \alpha & \sin \alpha & 0 \\ \cos \beta & \sin \beta & 0 \end{array} \\ &=(\cos x \sin \beta-\cos \beta \sin \alpha) \mathbf{k} \end{aligned} $$ and, from Eqs. $1-14$ and $1-15$ $$ \mathbf{a} \times \mathbf{b}=-\sin (\alpha-\beta) \mathbf{k} $$ Thus $$ \sin (\alpha-\beta)=\sin \alpha \cos \beta-\cos \alpha \sin \beta $$ Setting again $\beta^{t}=-\beta$ $$ \sin \left(\alpha+\beta^{\prime}\right)=\sin \alpha \cos \beta^{\prime}+\cos \alpha \sin \beta^{t} $$

Consider two unit vectors a and b, as shown in Fig. 1-19.
Find the trigonometric relations for the sine and cosine of the sum and diference of two angles from the values of $\mathbf{a} \cdot \mathbf{b}$ and $\mathbf{a} \times \mathbf{b}$.
Solution: First,
$$
\begin{aligned}
&\mathbf{a}=\cos \alpha \mathbf{i}+\sin \alpha \mathbf{j} \\
&\mathbf{b}=\cos \beta \mathbf{i}+\sin \beta \mathbf{j}
\end{aligned}
$$
Now we can write the product $\mathbf{a} \cdot \mathbf{b}$ in two different ways. From Eq. 1-11,
$$
\mathbf{a} \cdot \mathbf{b}=\cos \alpha \cos \beta+\sin \alpha \sin \beta
$$
and, from Eq. 1-5,
$$
\mathbf{a} \cdot \mathbf{b}=\cos (\alpha-\beta)
$$
Thus
$$
\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta
$$
If we set $\beta^{\prime}=-\beta$
$$
\cos \left(\alpha+\beta^{\prime}\right)=\cos \alpha \cos \beta^{\prime}-\sin \alpha \sin \beta^{\prime}
$$
Similarly, from Eq. 1-22,
$$
\begin{aligned}
\mathbf{a} \times \mathbf{b}=& \mid \begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\cos \alpha & \sin \alpha & 0 \\
\cos \beta & \sin \beta & 0
\end{array} \\
&=(\cos x \sin \beta-\cos \beta \sin \alpha) \mathbf{k}
\end{aligned}
$$
and, from Eqs. $1-14$ and $1-15$
$$
\mathbf{a} \times \mathbf{b}=-\sin (\alpha-\beta) \mathbf{k}
$$
Thus
$$
\sin (\alpha-\beta)=\sin \alpha \cos \beta-\cos \alpha \sin \beta
$$
Setting again $\beta^{t}=-\beta$
$$
\sin \left(\alpha+\beta^{\prime}\right)=\sin \alpha \cos \beta^{\prime}+\cos \alpha \sin \beta^{t}
$$

Key Concepts

Unit Vectors and Their Coordinate Representations

Unit vectors in a plane are often expressed in terms of cosine and sine components corresponding to an angle measured from a reference axis. This representation facilitates the translation between geometric angles and algebraic expressions, allowing one to use trigonometric functions to describe the orientation of the vectors.

Dot Product and Its Relation to Cosine

The dot product of two vectors offers a measure of how aligned the vectors are, and for unit vectors, it directly equals the cosine of the angle between them. This property is used to derive identities such as the cosine of the difference between two angles by equating the algebraic form of the dot product with the geometric interpretation in terms of angle.

Cross Product and Its Relation to Sine

The cross product of two vectors results in a vector whose magnitude is the product of the magnitudes of the original vectors and the sine of the angle between them. In the context of unit vectors, the magnitude of the cross product directly provides the sine of the angle between the vectors. This concept is essential for establishing the sine relation in the trigonometric identities.

Trigonometric Sum and Difference Formulas

The trigonometric sum and difference formulas are identities that express sine and cosine of sums or differences of angles in terms of the sines and cosines of the individual angles. These formulas are derived both from algebraic manipulations of the cosine and sine functions and from geometric interpretations using the dot and cross products of unit vectors.

Handling of Negative Angles

Using the even-odd properties of cosine and sine, negative angles can be incorporated into the trigonometric identities. This leads to transformations where cosine remains unchanged while sine changes sign, allowing derivations of formulas for both the sum and difference of angles by considering the angle sign appropriately.

Transcript

Please give Ace some feedback