Trigonometry Questions and Answers

Answer: B

The value of θ is 60°.

Check the options in the left-hand side. At 60°:

\[ \begin{aligned} \sin 60^{\circ} - \cos 60^{\circ} &= \frac{\sqrt{3}}{2} - \frac{1}{2} \\ &= \frac{\sqrt{3} - 1}{2} \end{aligned} \]

That matches. At 45° the difference is 0, at 90° it is 1, and at 75° it is \(\sin 75^{\circ} - \cos 75^{\circ} = \tfrac{\sqrt{2}}{2} \approx 0.71\), whereas \(\tfrac{\sqrt{3}-1}{2} \approx 0.37\).

Answer: C

The value is \(\dfrac{2x^{2}}{y^{2} - x^{2}}\).

From \(\sin\theta = x/y\), treat \(x\) as the opposite side and \(y\) as the hypotenuse. The adjacent side is then \(\sqrt{y^{2} - x^{2}}\):

\[ \begin{aligned} \tan\theta &= \frac{x}{\sqrt{y^{2} - x^{2}}} \\ 2\tan^{2}\theta &= \frac{2x^{2}}{y^{2} - x^{2}} \end{aligned} \]

The stem's '2.' is a multiplication dot. Options A and B are \(\csc\theta\) and \(\sin\theta\) themselves, and option D has the wrong power and the wrong sign in the denominator.

Answer: B

sec A is 17/15.

With the right angle at C, the hypotenuse is AB:

\[ AB = \sqrt{15^{2} + 8^{2}} = \sqrt{289} = 17 \]

For angle A, the adjacent side is AC = 15, so \(\cos A = 15/17\) and \(\sec A = 17/15\). 15/17 is cos A, the reciprocal of the answer, and 17/5 and 5/17 do not use the sides of this triangle at all.

Answer: B

The false option is \(\cos A = \sqrt{2}\).

\(\cot A = 1\) means \(A = 45^{\circ}\), so \(\sin A = \cos A = \frac{1}{\sqrt{2}}\). The cosine of an angle can never exceed 1, so \(\sqrt{2} \approx 1.414\) is impossible.

The other three are true: \(\sin A = \frac{1}{\sqrt{2}}\); \(\tan A + \cot A = 1 + 1 = 2\); and \(\cos A\,\text{cosec}\,A = \cos A/\sin A = \cot A = 1\).

Answer: D

The statement that is not true is \(\sec C = \sqrt{29}/2\).

With the right angle at B, \(\sin A = BC/AC = 5/\sqrt{29}\). Take BC = 5 and AC = \(\sqrt{29}\); then

\[ AB = \sqrt{29 - 25} = 2 \]

So \(\cos A = 2/\sqrt{29}\), \(\tan C = AB/BC = 2/5\) and \(\cos C = BC/AC = 5/\sqrt{29}\), all true. But \(\sec C = AC/BC = \sqrt{29}/5\), not \(\sqrt{29}/2\); the false option uses AB in place of BC.

A quick check: \(\sin^2 A + \cos^2 A = 25/29 + 4/29 = 1\), confirming sides 2, 5 and \(\sqrt{29}\). Since A and C are complementary, \(\cos C = \sin A\), which is why option C is true.

Answer: D

The value is 1.

tan P = 1 means P = 45°. Then sin P = cos P = \(\frac{1}{\sqrt{2}}\), so

\[ \begin{aligned} 2\sin P\cos P &= 2 \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}} \\ &= 1 \end{aligned} \]

The same follows from the identity 2 sin P cos P = sin 2P = sin 90° = 1. The value \(1/\sqrt{2}\) is sin P alone, \(\sqrt{2}\) is sin P + cos P, and 2 doubles the correct answer.

Answer: B

The value of \(\sec\theta\) is \(\frac{5}{3}\).

From \(3\sin\theta = 4\cos\theta\), \(\tan\theta = \frac{4}{3}\). Take a right triangle with opposite side 4 and adjacent side 3; its hypotenuse is 5:

\[ \begin{aligned} \sec\theta &= \frac{\text{hypotenuse}}{\text{adjacent}} \\ &= \frac{5}{3} \end{aligned} \]

\(\frac{5}{4}\) is \(\csc\theta\), \(\frac{4}{3}\) is \(\tan\theta\) and \(\frac{3}{4}\) is \(\cot\theta\), the usual ways of mixing up the ratios.

Answer: C

The value is 49/64.

Multiply out the brackets and use sin²θ + cos²θ = 1:

\[ \begin{aligned} \frac{(1+\sin\theta)(1-\sin\theta)}{(1+\cos\theta)(1-\cos\theta)} &= \frac{1-\sin^{2}\theta}{1-\cos^{2}\theta} \\ &= \frac{\cos^{2}\theta}{\sin^{2}\theta} = \cot^{2}\theta \end{aligned} \]

\[ \cot^{2}\theta = (7/8)^{2} = \frac{49}{64} \]

49/58 has the right numerator but a wrong denominator, and 21/24 and 35/48 do not come from squaring 7/8.

Answer: D

The value is \((2 - e^{2})^{3/2}\).

Reading the expression as \(\sec\alpha + \tan^{3}\alpha \cdot \csc\alpha\), first simplify the second term:

\[ \begin{aligned} \tan^{3}\alpha \csc\alpha &= \tan^{2}\alpha \cdot \frac{\sin\alpha}{\cos\alpha} \cdot \frac{1}{\sin\alpha} \\ &= \tan^{2}\alpha \sec\alpha \end{aligned} \]

so the whole expression is

\[ \begin{aligned} \sec\alpha(1 + \tan^{2}\alpha) &= \sec^{3}\alpha \\ &= (1 + \tan^{2}\alpha)^{3/2} \\ &= (2 - e^{2})^{3/2} \end{aligned} \]

\((2 - e^{2})^{1/2}\) is \(\sec\alpha\) alone, the first term without the second. The options with \(2 + e^{2}\) get the sign of \(e^{2}\) wrong when adding 1 to \(1 - e^{2}\).

Answer: A

45 degrees is 0.785 radian.

Since \(180^\circ = \pi\) radians:

\[ \begin{aligned} 45^\circ &= 45 \times \frac{\pi}{180} \\ &= \frac{\pi}{4} \approx 0.785 \end{aligned} \]

The other options are the radian values of other common angles: 1.57 is \(\frac{\pi}{2}\) (90°), 1.05 is \(\frac{\pi}{3}\) (60°) and 6.28 is \(2\pi\) (360°).