📝 Chapter 7: Trig Ratios & Sine/Cosine Laws

State the exact value of \( \tan 60° \).

In which quadrants is \( \sin\theta < 0 \)?

Find both values of \( \theta \) (between 0° and 360°) such that \( \cos\theta = -\frac{\sqrt{3}}{2} \).

In \( \triangle PQR \), \( p = 14, q = 8, R = 35° \). Find \( r \) using the cosine law.

In \( \triangle ABC \), \( A = 50°, B = 70°, c = 12 \). Find \( a \) using the sine law (round to 1 decimal).

In a right triangle, opposite = 9, adjacent = 12. Find \( \sec\theta \).

In \( \triangle ABC \): \( a = 13, b = 17, A = 38° \). Determine if there are 0, 1, or 2 triangles satisfying these conditions. If 2, find both values of \( B \).

A point \( (-4, 3) \) lies on the terminal arm of an angle in standard position. Find \( r \), \( \sin\theta, \cos\theta, \tan\theta \), and the principal angle \( \theta \).

Prove that for any acute angle \( \theta \), \( \sin^2\theta + \cos^2\theta = 1 \). Use a right triangle with hypotenuse \( h \).

Explain when to use the sine law and when to use the cosine law. Provide one example for each.

Use the CAST mnemonic to explain why an angle of 200° has a negative sine and a negative cosine. Find the reference angle and the exact ratios for \( \sin 210° \) and \( \cos 210° \).

Why does the SSA configuration sometimes produce two triangles? Use a labelled diagram in your explanation.

Compare the primary trig ratios with their reciprocals. Which is sec? csc? cot? Why are they useful?

A surveyor stands 80 m from the base of a building. The angle of elevation to the top of the building is 32°. The angle of elevation to a TV antenna on top of the building is 38°. Find the height of the antenna alone.

Two ships leave a port at the same time. Ship A travels at 20 km/h on a bearing of N30°E. Ship B travels at 25 km/h on a bearing of S70°E. After 3 hours, how far apart are the ships?

A triangular plot of land has sides 240 m, 300 m, and 400 m. Find all three angles, then compute the area of the plot using \( A = \frac{1}{2}ab\sin C \).