Question

question 9
select all true statements.
image of a right triangle with right angle at b, ab=5, bc=12, angle at a is α, angle at c is θ
□ a. ac is √119
□ b. ac is 13 units
□ c. cos(θ) = 5/12
□ d. sin(α) = 12/13
□ e. θ = tan⁻¹(5/12)

Explanation:

Step1: Calculate AC length

In right triangle \(ABC\) (right - angled at \(B\)), by Pythagorean theorem \(AC^{2}=AB^{2}+BC^{2}\). Given \(AB = 5\), \(BC=12\), so \(AC^{2}=5^{2}+12^{2}=25 + 144=169\), then \(AC=\sqrt{169} = 13\). So option a is wrong, option b is correct.

Step2: Analyze \(\cos(\theta)\)

For angle \(\theta\) at \(C\), in right triangle \(ABC\), \(\cos(\theta)=\frac{adjacent}{hypotenuse}=\frac{BC}{AC}=\frac{12}{13}\), not \(\frac{5}{12}\). So option c is wrong.

Step3: Analyze \(\sin(\alpha)\)

For angle \(\alpha\) at \(A\), \(\sin(\alpha)=\frac{opposite}{hypotenuse}=\frac{BC}{AC}=\frac{12}{13}\). So option d is correct.

Step4: Analyze \(\theta=\tan^{- 1}(\frac{5}{12})\)

For angle \(\theta\) at \(C\), \(\tan(\theta)=\frac{opposite}{adjacent}=\frac{AB}{BC}=\frac{5}{12}\), so \(\theta=\tan^{-1}(\frac{5}{12})\). Option e is correct.

Answer:

b. \(AC\) is 13 units, d. \(\sin(\alpha)=\frac{12}{13}\), e. \(\theta=\tan^{-1}(\frac{5}{12})\)