**Hint:** Describe trigonometry in order to study the relationship between side and angle of a triangle.

**Question.1.** Which of the following completes the statement below?

The adjacent and ______ of right triangle depends on the \angle x being referred to in that triangle and for the complementary angle (90 − 𝑥)° & of that triangle, the adjacent and opposite are ___________.

(a) opposite side, reversed

(b) opposite side, same

(c) hypotenuse, reversed

(d) hypotenuse, same

**Question.2.** Considering the diagram below. Which of the following statements is true?

(a) Side AC is adjacent to \angle y in triangle ADC and triangle ABC

(b) Side AC is adjacent to \angle y in triangle ADC and side BC is adjacent to \angle y in triangle ABC

(c) Side DC is adjacent to \angle y in triangle ADC and side AC is adjacent to \angle y in triangle ABC

(d) Side AC is adjacent to \angle y in triangle ADC and side DC is adjacent to \angle y in triangle ABC

**Ans.1.** (a) opposite side, reversed**Ans.2.** (c) Side DC is adjacent to \angle y in triangle ADC and side AC is adjacent to \angle y in triangle ABC

**Hint:** Define and distinguish various trigonometric ratios in order to describe and verify sine, cosine, tangent, cosecant, secant, cotangent of an angle.

**Question.3.** Consider the triangle shown below. What are the values of tan θ, cosec θ and sec θ?

(a) tan θ = \frac{8}{15}, cosec θ = \frac{17}{15}, sec θ = \frac{17}{8}

(b) tan θ = \frac{8}{15}, cosec θ = \frac{17}{8}, sec θ = \frac{17}{15}

(c) tan θ = \frac{17}{15}, cosec θ = \frac{8}{15}, sec θ = \frac{17}{8}

(d) tan θ = \frac{8}{15}, cosec θ = \frac{17}{15}, sec θ = \frac{8}{17}

**Question.4.** Observe the figure shown. Which of these is the value of cos θ?

(a) \frac{1}{2}

(b) \frac{2}{1}

(c) \frac{2\sqrt{3}}{3}

(d) \frac{3\sqrt{3}}{2}

**Ans.3.** (b) tan θ = \frac{8}{15}, cosec θ = \frac{17}{8}, sec θ = \frac{17}{15}**Ans.4.** (a) \frac{1}{2}

**Hint:** Use given trigonometric ratio(s) in order to find and verify other trigonometric ratios/angles of the triangle.

**Question.5.** If sin θ = \frac{7}{\sqrt{85}}, what are the values of tan θ, cos θ and cosec θ?

(a) tan θ = \frac{7}{6}, cos θ =\frac{6}{\sqrt{85}} and cosec θ = \frac{\sqrt{85}}{7}

(b) tan θ = \frac{6}{7}, cos θ =\frac{7}{\sqrt{85}} and cosec θ = \frac{\sqrt{85}}{7}

(c) tan θ = \frac{7}{6}, cos θ =\frac{7}{\sqrt{85}} and cosec θ = \frac{\sqrt{85}}{7}

(d) tan θ = \frac{6}{7}, cos θ =\frac{6}{\sqrt{85}} and cosec θ = \frac{\sqrt{85}}{6}

**Question.6.** The two legs AB and BC of right triangle ABC are in a ratio 1:3. What will be the value of sin C?

(a) \frac{1}{\sqrt{10}}

(b) \frac{3}{\sqrt{10}}

(c) \frac{1}{3}

(d) \frac{1}{2}

**Ans.5.** (a) tan θ = \frac{7}{6}, cos θ =\frac{6}{\sqrt{85}} and cosec θ = \frac{\sqrt{85}}{7}**Ans.6.** (a) \frac{1}{\sqrt{10}}

**Hint:** Compute the trigonometric ratio of 0°, 30°, 45°, 60°, 90° in order to know and apply the value of specific angles.

**Question.7.** What is the value of \frac{3-\sin^{2}60°}{\tan 30° \tan 60°}?

(a) 1\frac{1}{4}

(b) 2\frac{1}{4}

(c) 2\frac{1}{2}

(d) 2\frac{3}{4}

**Question.8.** The value of \frac{4-\sin^{2}45°}{\cot k \tan 60°} is 3.5.

What is the value of k?

(a) 30°

(b) 60°

(c) 45°

(d) 90°

**Ans.7.** (b) 2\frac{1}{4}**Ans.8.** (b) 60°

**Hint:** Compute the trigonometric ratio of complimentary angles in order to apply the values in mathematical problems.

**Question.9.** What is the value of \frac{5 \csc 55°}{\sec 35°}?

(a) \frac{5}{2}

(b) \frac{15}{2}

(c) 5

(d) 15

**Question.10.** If \sec 2x = \frac{1}{\sin (x-36)°}, where 2x is an acute angle, what is the value of x?

(a) 36

(b) 42

(c) 48

(d) 52

**Ans.9.** (b) \frac{15}{2}**Ans.10.** (b) 42

**Hint:** Compute and apply trigonometric identities in order to simplify and solve mathematical problems.

**Question.11.** Which of these is equivalent to \frac{2 \tan x (\sec^{2}x-1)}{\cos^{3}x}?

(a) 2 \tan^{3}x \csc x

(b) 2 \tan^{3}x \sec^{3} x

(c) 2 \tan^{3}x \csc^{3} x

(d) 2 \cot^{3}x \sec^{3} x

**Question.12.** Which of the following option makes the statement below true?

\frac{\frac{1}{\sec x}+\sec x}{\cos^{2}x-1-\tan^{2}x} = ?

(a) −cosec x cot x

(b) − sec x cot x

(c) −cosec x tan x

(d) − sec x tan x

**Ans.11.** (b) 2 \tan^{3}x \sec^{3} x**Ans.12.** (a) −cosec x cot x