Question 5 and 6, Exercise 8.1

Solutions of Question 5 and 6 of Exercise 8.1 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

For $\sin \alpha=\dfrac{4}{5}, \tan \beta=-\dfrac{5}{12}$ with terminal side of an angles in QII, find $\cos (\alpha+\beta)$ and $\cos (\alpha-\beta)$.

Solution.

Given: $\sin \alpha=\dfrac{4}{5}$, $\alpha$ is in QII and $\tan \beta=-\dfrac{5}{12}$, $\beta$ is in QII.

We have an identity: $$\cos \alpha=\pm \sqrt{1-\sin^2\alpha}.$$ As $\alpha$ lies in QII and $\cos$ is -ive in QII, $$\begin{align*}\cos\alpha &=-\sqrt{1-\sin^2\alpha}\\ &=-\sqrt{1-{{\left(-\frac{4}{5} \right)}^{2}}}\\ &=-\sqrt{1-\dfrac{16}{25}}=\sqrt{\dfrac{9}{25}}\\ \Rightarrow \quad \cos \alpha&=-\dfrac{3}{5}.\end{align*}$$ Also $$\sec \beta=\pm\sqrt{1+\tan^2\beta}.$$ As $\beta$ lies in QII and $\sec$ is -ive in QII, $$\begin{align*}\sec \beta & =-\sqrt{1+\tan^2\beta} \\ &=-\sqrt{1+{{\left(-\dfrac{5}{12} \right)}^{2}}}\\ &=-\sqrt{1+\dfrac{25}{144}} \\ & =-\sqrt{\dfrac{169}{144}} =-\dfrac{13}{12} \end{align*}$$

$$\begin{align*} \Rightarrow \quad \cos \beta=\frac{1}{\sec \beta}=-\frac{12}{13}.\end{align*}$$ This gives \begin{align*} \frac{\sin\beta}{\cos\beta} & = \tan\beta \\ \implies \sin\beta & = \tan\beta \cos\beta \\ & = \left(-\frac{5}{12} \right) \left(-\frac{12}{13} \right) \\ \implies \sin\beta & = \frac{5}{13}. \end{align*} Now $$\begin{align*}\cos (\alpha+\beta )&=\cos \alpha\cos \beta-\sin \alpha\sin \beta \\ &=\left(-\frac{3}{5}\right)\left(-\frac{12}{13}\right)-\left(\frac{4}{5}\right)\left(\frac{5}{13}\right)\\ &=\dfrac{36}{65}-\frac{20}{65} =\frac{16}{65} \end{align*}$$ Also $$\begin{align*}\cos (\alpha-\beta )&=\cos \alpha\cos \beta+\sin \alpha\sin \beta \\ &=\left(-\frac{3}{5}\right)\left(-\frac{12}{13}\right)+\left(\frac{4}{5}\right)\left(\frac{5}{13}\right)\\ &=\dfrac{36}{65}+\frac{20}{65} =\frac{56}{65} \end{align*}$$ Hence $$\begin{align*}\boxed{\cos(\alpha+\beta)=\frac{16}{65}} \text{ and } \boxed{\cos(\alpha-\beta)=\frac{56}{65}}\end{align*}$$

For $\cos \alpha=-\dfrac{7}{25}$ with terminal side of $\alpha$ in QII and $\cot \beta=\dfrac{15}{8}$ with terminal side of $\beta$ in QIII, find
(i) $\sin(\alpha-\beta)$ (ii) $\cos(\alpha-\beta)$ (iii) $\tan(\alpha-\beta)$.

Solution.

Given: $\cos\alpha=-\dfrac{7}{25}$, $\alpha$ is in QII and $\cot\beta=\dfrac{15}{8}$, $\beta$ is in QIII.

We have an identity: $$\sin \alpha=\pm \sqrt{1-\cos^2\alpha}.$$ As $\alpha$ lies in QII and $\sin$ is +ive in QII, \begin{align*}\sin\alpha &=\sqrt{1-\cos^2\alpha}\\ &=\sqrt{1-{{\left(-\frac{7}{25} \right)}^{2}}}\\ &=\sqrt{1-\dfrac{49}{625}}=\sqrt{\dfrac{576}{625}}\\ \implies \sin \alpha&=\dfrac{24}{25}.\end{align*} Also $$\csc \beta=\pm\sqrt{1+\cot^2\beta}.$$ As $\beta$ lies in QIII and $\csc$ is -ive in QIII, $$\begin{align*}\csc \beta & =-\sqrt{1+\cot^2\beta} \\ &=-\sqrt{1+{{\left(-\dfrac{15}{8} \right)}^{2}}}\\ &=-\sqrt{1+\dfrac{225}{64}} \\ & =-\sqrt{\dfrac{289}{64}} =-\dfrac{17}{8} \end{align*}$$

$$\begin{align*} \Rightarrow \quad \sin \beta=\frac{1}{\csc \beta}=-\frac{8}{17}.\end{align*}$$ This gives \begin{align*} \frac{\cos\beta}{\sin\beta} & = \cot\beta \\ \implies \cos\beta & = \cot\beta \sin\beta \\ & = \left(\frac{15}{8} \right) \left(-\frac{8}{17} \right) \\ \implies \cos\beta & = -\frac{15}{17}. \end{align*}

(i) $\sin(\alpha-\beta)$ $$\begin{align*}\sin (\alpha-\beta )&=\sin \alpha\cos \beta-\cos \alpha\sin \beta \\ &=\left(\frac{24}{25}\right)\left(-\frac{15}{17}\right)-\left(-\frac{7}{25}\right)\left(-\frac{8}{17}\right)\\ &=-\dfrac{72}{85}-\frac{56}{425} =-\frac{416}{425} \end{align*}$$

(ii) $\cos(\alpha-\beta)$ $$\begin{align*}\cos (\alpha-\beta )&=\cos \alpha\cos \beta+\sin \alpha\sin \beta \\ &=\left(-\frac{7}{25}\right)\left(-\frac{15}{17}\right)+\left(\frac{24}{25}\right)\left(-\frac{8}{17}\right)\\ &=\dfrac{21}{85}-\frac{192}{425} =-\frac{87}{425} \end{align*}$$

(iii) $\tan(\alpha-\beta)$ $$\begin{align*}\tan(\alpha-\beta) & = \frac{\sin(\alpha-\beta)}{\cos (\alpha-\beta)}\\ &= \frac{-416/425}{-87/425} = \frac{416}{87} \end{align*}$$

Hence
$\boxed{\sin(\alpha-\beta)=-\frac{416}{425}}$, $\boxed{\cos(\alpha-\beta)=-\frac{87}{425}}$, $\boxed{\tan(\alpha-\beta)=\frac{416}{87}}$.

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