Trigonometry Questions and Answers

Answer: B

We use the complementary angle identity \(\tan(90° - \theta) = \cot \theta\), and the fact that \(\tan \theta \cdot \cot \theta = 1\).

The expression can be paired up: \((\tan 1° \tan 89°) (\tan 2° \tan 88°) ...\)

Since \(\tan 89° = \tan(90°-1°) = \cot 1°\), the first pair is \(\tan 1° \cot 1° = 1\).

All pairs will multiply to 1. The middle term is \(\tan 45°\), which is also 1.

The final product is 1.

Answer: B

From the given equation, we can write:

\(\sin \theta = 1 - \sin^2 \theta\)

Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we know that \(1 - \sin^2 \theta = \cos^2 \theta\).

Therefore, we have \(\sin \theta = \cos^2 \theta\).

Now we evaluate the expression \(\cos^2 \theta + \cos^4 \theta\). We can write \(\cos^4 \theta\) as \((\cos^2 \theta)^2\).

Substituting \(\cos^2 \theta = \sin \theta\), the expression becomes:

\(\sin \theta + (\sin \theta)^2 = \sin \theta + \sin^2 \theta\)

From the original given equation, we know that \(\sin \theta + \sin^2 \theta = 1\).

So, the value of the expression is 1.

Answer: A

This expression is in the form of the tangent subtraction formula: \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\).

Here, A = 60° and B = 30°.

So, the expression is equal to \(\tan(60° - 30°) = \tan 30°\).

Answer: D

Let the length of the ladder be L (the hypotenuse) and the distance from the wall be D (the adjacent side).

We use the cosine function: \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\).

\(\cos 60° = \frac{4.6}{L}\)

We know \(\cos 60° = 1/2\).

\(\frac{1}{2} = \frac{4.6}{L}\)

L = 4.6 × 2 = 9.2 meters.

Answer: A

We can write 75° as the sum of two standard angles, 45° + 30°.

Using the sine addition formula: \(\sin(A+B) = \sin A \cos B + \cos A \sin B\).

\(\sin(45°+30°) = \sin 45° \cos 30° + \cos 45° \sin 30°\)

= \((\frac{1}{\sqrt{2}})(\frac{\sqrt{3}}{2}) + (\frac{1}{\sqrt{2}})(\frac{1}{2})\)

= \(\frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{3}+1}{2\sqrt{2}}\).

Answer: D

If \(\sin \theta = \cos \theta\), dividing both sides by \(\cos \theta\) gives \(\tan \theta = 1\). This means \(\theta = 45°\).

Now we evaluate the expression: \(2\tan 45° + \cos^2 45°\).

= \(2(1) + (\frac{1}{\sqrt{2}})^2\)

= \(2 + \frac{1}{2} = \frac{5}{2}\).

Answer: B

If \(\tan A = 1\), then A = 45°.

The expression \(2 \sin A \cos A\) is the double angle identity for sine, \(\sin(2A)\).

So, we need to find \(\sin(2 \times 45°) = \sin(90°)\).

The value of \(\sin 90°\) is 1.

Answer: B

Take out the common factor and use the identity.

\[ 9(\sec^{2}A - \tan^{2}A) = 9 \times 1 = 9 \]

The identity \( \sec^{2}A - \tan^{2}A = 1 \) comes from dividing \( \sin^{2}A + \cos^{2}A = 1 \) through by \( \cos^{2}A \).

Answer: B

\[ \tan 60^{\circ} = \frac{h}{20} \]

\[ h = 20\sqrt{3} \approx 34.6 \text{ m} \]

At 60° the tower is \( \sqrt{3} \) times the ground distance - taller than 20 m, which rules out two of the options at once.

Answer: A

This expression is in the form of the cosine addition formula: \(\cos(A+B) = \cos A \cos B - \sin A \sin B\).

Here, A = 38° and B = 52°.

The expression is equal to \(\cos(38° + 52°) = \cos(90°)\).

The value of \(\cos 90°\) is 0.