Evaluate the following : (i) (ii) (iii) (iv) (v)

Understand the Question To evaluate trigonometric expressions, substitute standard values for specific angles () and simplify algebraically. Key standard trigonometric values: Reciprocal identities: , , and . Part (i) Evaluate Step 1 · Substitute Values and Evaluate Substitute , , , and Answer > (i) Part (ii) Evaluate Step 1 · Substitute Values and Evaluate Substitute , , and Answer > (ii) Part (iii) Evaluate Step 1 · Substitute Values and Simplify Substitute , , and Step 2 · Rationalise the Denominator Rationalising the denominator by multiplying numerator and denominator by Answer > (iii) Part (iv) Evaluate Step 1 · Substitute Values and Simplify Substitute , , , , , and Step 2 · Rationalise the Denominator Rationalising the denominator by multiplying numerator and denominator by Answer > (iv) Part (v) Evaluate Step 1 · Evaluate Numerator and Denominator Substitute , , , , and Numerator: Denominator: Divide Numerator by Denominator: Answer > (v) Common Mistakes Value Swapping: Confusing values of complementary angles, such as writing instead of , or instead of . Rationalisation Errors: When rationalising , rearrange terms to first to keep the denominator positive and avoid negative sign errors. Identity Shortcut: In part (v), the denominator simplifies directly to using the fundamental identity .

Introduction to Trigonometry | Exercise 8.2

Question 1

Evaluate the following :

(i) $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$

(ii) $2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ$

(iii) $\dfrac{\cos 45^\circ}{\sec 30^\circ + \csc 30^\circ}$

(iv) $\dfrac{\sin 30^\circ + \tan 45^\circ - \csc 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$

(v) $\dfrac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$

Solution

Understand the Question

To evaluate trigonometric expressions, substitute standard values for specific angles ( $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$ ) and simplify algebraically.
Key standard trigonometric values:

\begin{array}{|c|c|c|c|} \hline \text{Angle} & \sin & \cos & \tan \\ \hline 0^\circ & 0 & 1 & 0 \\ \hline 30^\circ & \dfrac{1}{2} & \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{3}} \\ \hline 45^\circ & \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} & 1 \\ \hline 60^\circ & \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} & \sqrt{3} \\ \hline 90^\circ & 1 & 0 & \text{undefined} \\ \hline \end{array}

Reciprocal identities: $\sec \theta = \dfrac{1}{\cos \theta}$ , $\csc \theta = \dfrac{1}{\sin \theta}$ , and $\cot \theta = \dfrac{1}{\tan \theta}$ .

(i) Evaluate $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$

Step 1 · Substitute Values and Evaluate

Diagram 1

Substitute $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$ , $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$ , $\sin 30^\circ = \dfrac{1}{2}$ , and $\cos 60^\circ = \dfrac{1}{2}$

\begin{aligned} \sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ &= \left(\dfrac{\sqrt{3}}{2}\right) \left(\dfrac{\sqrt{3}}{2}\right) + \left(\dfrac{1}{2}\right) \left(\dfrac{1}{2}\right) \\[0.6em] &= \dfrac{3}{4} + \dfrac{1}{4} \\[0.6em] &= \dfrac{4}{4} \\[0.6em] &= 1 \end{aligned}

Answer

(i) $1$

(ii) Evaluate $2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ$

Step 1 · Substitute Values and Evaluate

Substitute $\tan 45^\circ = 1$ , $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$ , and $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$

\begin{aligned} 2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ &= 2(1)^2 + \left(\dfrac{\sqrt{3}}{2}\right)^2 - \left(\dfrac{\sqrt{3}}{2}\right)^2 \\[0.6em] &= 2(1) + \dfrac{3}{4} - \dfrac{3}{4} \\[0.6em] &= 2 + 0 \\[0.6em] &= 2 \end{aligned}

Answer

(ii) $2$

(iii) Evaluate $\dfrac{\cos 45^\circ}{\sec 30^\circ + \csc 30^\circ}$

Step 1 · Substitute Values and Simplify

Substitute $\cos 45^\circ = \dfrac{1}{\sqrt{2}}$ , $\sec 30^\circ = \dfrac{2}{\sqrt{3}}$ , and $\csc 30^\circ = 2$

\begin{aligned} \dfrac{\cos 45^\circ}{\sec 30^\circ + \csc 30^\circ} &= \dfrac{\dfrac{1}{\sqrt{2}}}{\dfrac{2}{\sqrt{3}} + 2} \\[1.1em] &= \dfrac{\dfrac{1}{\sqrt{2}}}{\dfrac{2 + 2\sqrt{3}}{\sqrt{3}}} \\[1.1em] &= \dfrac{1}{\sqrt{2}} \times \dfrac{\sqrt{3}}{2 + 2\sqrt{3}} \\[0.6em] &= \dfrac{\sqrt{3}}{2\sqrt{2} + 2\sqrt{6}} \end{aligned}

Step 2 · Rationalise the Denominator

Rationalising the denominator by multiplying numerator and denominator by $(2\sqrt{6} - 2\sqrt{2})$

\begin{aligned} \dfrac{\sqrt{3}}{2\sqrt{2} + 2\sqrt{6}} &= \dfrac{\sqrt{3}}{2\sqrt{6} + 2\sqrt{2}} \times \dfrac{2\sqrt{6} - 2\sqrt{2}}{2\sqrt{6} - 2\sqrt{2}} \\[0.6em] &= \dfrac{2\sqrt{18} - 2\sqrt{6}}{(2\sqrt{6})^2 - (2\sqrt{2})^2} \\[0.6em] &= \dfrac{2 \times 3\sqrt{2} - 2\sqrt{6}}{24 - 8} \\[0.6em] &= \dfrac{6\sqrt{2} - 2\sqrt{6}}{16} \\[0.6em] &= \dfrac{2(3\sqrt{2} - \sqrt{6})}{16} \\[0.6em] &= \dfrac{3\sqrt{2} - \sqrt{6}}{8} \end{aligned}

Answer

(iii) $\dfrac{3\sqrt{2} - \sqrt{6}}{8}$

(iv) Evaluate $\dfrac{\sin 30^\circ + \tan 45^\circ - \csc 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$

Step 1 · Substitute Values and Simplify

Substitute $\sin 30^\circ = \dfrac{1}{2}$ , $\tan 45^\circ = 1$ , $\csc 60^\circ = \dfrac{2}{\sqrt{3}}$ , $\sec 30^\circ = \dfrac{2}{\sqrt{3}}$ , $\cos 60^\circ = \dfrac{1}{2}$ , and $\cot 45^\circ = 1$

\begin{aligned} \dfrac{\sin 30^\circ + \tan 45^\circ - \csc 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ} &= \dfrac{\dfrac{1}{2} + 1 - \dfrac{2}{\sqrt{3}}}{\dfrac{2}{\sqrt{3}} + \dfrac{1}{2} + 1} \\[1.1em] &= \dfrac{\dfrac{3}{2} - \dfrac{2}{\sqrt{3}}}{\dfrac{2}{\sqrt{3}} + \dfrac{3}{2}} \\[1.1em] &= \dfrac{\dfrac{3\sqrt{3} - 4}{2\sqrt{3}}}{\dfrac{4 + 3\sqrt{3}}{2\sqrt{3}}} \\[1.1em] &= \dfrac{3\sqrt{3} - 4}{3\sqrt{3} + 4} \end{aligned}

Step 2 · Rationalise the Denominator

Rationalising the denominator by multiplying numerator and denominator by $(3\sqrt{3} - 4)$

\begin{aligned} \dfrac{3\sqrt{3} - 4}{3\sqrt{3} + 4} &= \dfrac{3\sqrt{3} - 4}{3\sqrt{3} + 4} \times \dfrac{3\sqrt{3} - 4}{3\sqrt{3} - 4} \\[0.6em] &= \dfrac{(3\sqrt{3})^2 - 2(3\sqrt{3})(4) + (4)^2}{(3\sqrt{3})^2 - (4)^2} \\[0.6em] &= \dfrac{27 - 24\sqrt{3} + 16}{27 - 16} \\[0.6em] &= \dfrac{43 - 24\sqrt{3}}{11} \end{aligned}

Answer

(iv) $\dfrac{43 - 24\sqrt{3}}{11}$

(v) Evaluate $\dfrac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$

Step 1 · Evaluate Numerator and Denominator

Substitute $\cos 60^\circ = \dfrac{1}{2}$ , $\sec 30^\circ = \dfrac{2}{\sqrt{3}}$ , $\tan 45^\circ = 1$ , $\sin 30^\circ = \dfrac{1}{2}$ , and $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$

Numerator:

\begin{aligned} 5 \left(\dfrac{1}{2}\right)^2 + 4 \left(\dfrac{2}{\sqrt{3}}\right)^2 - (1)^2 &= 5 \left(\dfrac{1}{4}\right) + 4 \left(\dfrac{4}{3}\right) - 1 \\[0.6em] &= \dfrac{5}{4} + \dfrac{16}{3} - 1 \\[0.6em] &= \dfrac{15}{12} + \dfrac{64}{12} - \dfrac{12}{12} \\[0.6em] &= \dfrac{15 + 64 - 12}{12} \\[0.6em] &= \dfrac{67}{12} \end{aligned}

Denominator:

\begin{aligned} \sin^2 30^\circ + \cos^2 30^\circ &= \left(\dfrac{1}{2}\right)^2 + \left(\dfrac{\sqrt{3}}{2}\right)^2 \\[0.6em] &= \dfrac{1}{4} + \dfrac{3}{4} \\[0.6em] &= \dfrac{4}{4} \\[0.6em] &= 1 \end{aligned}

Divide Numerator by Denominator: $\dfrac{\dfrac{67}{12}}{1} = \dfrac{67}{12}$

Answer

(v) $\dfrac{67}{12}$

Common Mistakes

Value Swapping: Confusing values of complementary angles, such as writing $\sin 60^\circ = \dfrac{1}{2}$ instead of $\dfrac{\sqrt{3}}{2}$ , or $\cos 60^\circ = \dfrac{\sqrt{3}}{2}$ instead of $\dfrac{1}{2}$ .
Rationalisation Errors: When rationalising $\dfrac{\sqrt{3}}{2\sqrt{2} + 2\sqrt{6}}$ , rearrange terms to $2\sqrt{6} + 2\sqrt{2}$ first to keep the denominator positive and avoid negative sign errors.
Identity Shortcut: In part (v), the denominator $\sin^2 30^\circ + \cos^2 30^\circ$ simplifies directly to $1$ using the fundamental identity $\sin^2\theta + \cos^2\theta = 1$ .

Question 1

More questions in Exercise 8.2