Question 1
In a right triangle, opposite = 5, hypotenuse = 13. \( \sin\theta = ? \)
Solution:
SOH: \( \sin\theta = \frac{\text{opp}}{\text{hyp}} = \frac{5}{13} \).
Question 2
Exact value of \( \sin 60° \):
Solution:
From the 30-60-90 reference triangle: \( \sin 60° = \frac{\sqrt{3}}{2} \).
Question 3
Exact value of \( \cos 45° \):
Solution:
45-45-90 triangle: \( \cos 45° = \frac{\sqrt{2}}{2} \).
Question 4
Exact value of \( \tan 30° \):
Solution:
\( \tan 30° = \frac{\sin 30°}{\cos 30°} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} \).
Question 5
In Quadrant III, which trig ratio is positive?
Solution:
CAST: Q3 → tan positive only.
Question 6
If \( \sin\theta = 0.5 \) and \( 0° \leq \theta \leq 360° \), the two values of \( \theta \) are \( 30° \) and:
Solution:
Sin positive in Q1 and Q2. Reference angle 30°. Q2: \( 180° - 30° = 150° \).
Question 7
If \( \cos\theta = -\tfrac{1}{2} \) and \( 0° \leq \theta \leq 360° \), the smaller of the two values is:
Solution:
Cos negative in Q2, Q3. Reference 60°. Q2: 180-60=120°; Q3: 180+60=240°. Smaller: 120°.
Question 8
In \( \triangle ABC \), \( a = 12, A = 35°, B = 80° \). Find \( b \) (round to one decimal).
Solution:
\( \frac{b}{\sin 80°} = \frac{12}{\sin 35°} \). \( b = \frac{12 \sin 80°}{\sin 35°} \approx \frac{11.818}{0.5736} \approx 20.6 \).
Question 9
In \( \triangle ABC \), \( a = 8, b = 15, c = 17 \). Find angle \( C \) (degrees, rounded).
Solution:
\( \cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{64 + 225 - 289}{240} = 0 \). So \( C = 90° \).
Question 10
In \( \triangle ABC \), \( a = 7, b = 9, C = 40° \). Find \( c \) (round to one decimal).
Solution:
\( c^2 = 49 + 81 - 2(7)(9)\cos 40° = 130 - 126(0.766) = 130 - 96.5 = 33.5 \). \( c \approx 5.8 \).
Question 11
For SSA with \( a = 10, b = 14, A = 30° \), the ambiguous case yields how many triangles?
Solution:
\( \sin B = \frac{14 \sin 30°}{10} = 0.7 \), so \( B \approx 44.4° \) or \( B \approx 135.6° \). Both valid.
Question 12
A reference angle is always:
Solution:
The reference angle is the acute angle between the terminal arm and the x-axis. It is always between 0° and 90°.