$\frac{\cos 140^{\circ}-\cos 100^{\circ}}{\sin 140^{\circ}-\sin 100^{\circ}}=$

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Algebra 2

Find trigonometric ratios using reference angles

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\frac{\cos 140^{\circ}-\cos 100^{\circ}}{\sin 140^{\circ}-\sin 100^{\circ}}=

Use Cosine Identity: Use the cosine of the supplementary angle identity. $\newline$ The cosine of a supplementary angle $180° - \theta$ is equal to the negative cosine of the angle $\cos(180° - \theta) = -\cos(\theta)$ . This means that $\cos(140°) = -\cos(40°)$ and $\cos(100°) = -\cos(80°)$ .
Use Sine Identity: Use the sine of the supplementary angle identity. $\newline$ The sine of a supplementary angle $180° - \theta$ is equal to the sine of the angle $\sin(180° - \theta) = \sin(\theta)$ . This means that $\sin(140°) = \sin(40°)$ and $\sin(100°) = \sin(80°)$ .
Substitute Values: Substitute the values from Step $1$ and Step $2$ into the original expression. $\newline$ The expression becomes $(-\cos(40°) + \cos(80°)) / (\sin(40°) - \sin(80°))$ .
Use Co-function Identities: Use the sine and cosine co-function identities. $\newline$ The co-function identities state that $\sin(\theta) = \cos(90° - \theta)$ and $\cos(\theta) = \sin(90° - \theta)$ . This means that $\cos(80°) = \sin(10°)$ and $\cos(40°) = \sin(50°)$ .
Substitute Values: Substitute the values from Step $4$ into the expression from Step $3$ . $\newline$ The expression now becomes $(-\sin(50^\circ) + \sin(10^\circ)) / (\sin(40^\circ) - \sin(80^\circ))$ .
Use Sine Identities: Use the sine difference and sum identities. $\newline$ The sine difference identity is $\sin(a) - \sin(b) = 2 \cdot \cos\left(\frac{a + b}{2}\right) \cdot \sin\left(\frac{a - b}{2}\right)$ . Applying this to both the numerator and the denominator, we get: $\newline$ Numerator: $2 \cdot \cos\left(\frac{50° + 10°}{2}\right) \cdot \sin\left(\frac{50° - 10°}{2}\right) = 2 \cdot \cos(30°) \cdot \sin(20°)$ $\newline$ Denominator: $2 \cdot \cos\left(\frac{40° + 80°}{2}\right) \cdot \sin\left(\frac{40° - 80°}{2}\right) = 2 \cdot \cos(60°) \cdot \sin(-20°)$
Simplify Expression: Simplify the expression. $\newline$ Since $\sin(-\theta) = -\sin(\theta)$ , the denominator becomes $-2 \times \cos(60^\circ) \times \sin(20^\circ)$ . The $2$ 's and $\sin(20^\circ)$ terms cancel out in the numerator and denominator, leaving us with: $\newline$ $\cos(30^\circ) / -\cos(60^\circ)$
Calculate Cosine Values: Calculate the exact values of $\cos(30°)$ and $\cos(60°)$ . $\newline$ $\cos(30°) = \sqrt{3}/2$ and $\cos(60°) = 1/2$ .
Substitute Values: Substitute the exact values into the simplified expression. $\newline$ The final expression is $(\sqrt{3}/2) / -(1/2)$ .
Simplify Final Expression: Simplify the final expression. $\newline$ When we divide by a fraction, it is the same as multiplying by its reciprocal. Therefore, the final answer is: $\newline$ $(\sqrt{3}/2) \times -(2/1) = -\sqrt{3}$