1. The problem asks to verify the truthfulness of several statements related to trigonometric functions and triangle properties based on a right triangle with sides 10 ft (adjacent), 30 ft (opposite), and hypotenuse unknown. 2. Recall the definitions of trigonometric functions for an acute angle $\theta$ in a right triangle: - $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ 3. Calculate the hypotenuse using the Pythagorean theorem: $$\text{hypotenuse} = \sqrt{10^2 + 30^2} = \sqrt{100 + 900} = \sqrt{1000} = 31.62$$ 4. Evaluate each statement: - Statement 1: True, by definition of cosine. - Statement 2: False, sine is opposite over hypotenuse, not hypotenuse over adjacent. - Statement 3: True, tangent is opposite over adjacent. - Statement 4: True, $\tan \theta = \frac{30}{10} = 3$, so $\frac{10}{30} = \frac{1}{3}$ is incorrect; statement says tan $\theta$ is $\frac{10}{30}$ which is false, so statement 4 is False. - Statement 5: False, since $\tan \theta = \frac{30}{10}$, $\theta = \tan^{-1}(3)$, not $\tan^{-1}(\frac{30}{10})$ which is the same but the statement is ambiguous; assuming it means $\tan^{-1}(\frac{30}{10})$ which is correct, so statement 5 is True. - Statement 6: True, hypotenuse calculated as 31.62. - Statement 7: False, formula $A = a^2 + b^2 - 2ab \cos C$ is the Law of Cosines, used to find a side length, not area. - Statement 8: True, $A = \frac{1}{2}ab \sin C$ is the area formula for two sides and included angle. - Statement 9: Calculate area using Heron's formula for sides 5,7,8: - Semi-perimeter $s = \frac{5+7+8}{2} = 10$ - Area $= \sqrt{s(s-5)(s-7)(s-8)} = \sqrt{10 \times 5 \times 3 \times 2} = \sqrt{300} = 17.32$ So statement 9 is True. - Statement 10: Area with sides 12,14 and included angle 40°: - $A = \frac{1}{2} \times 12 \times 14 \times \sin 40^\circ$ - $\sin 40^\circ \approx 0.6428$ - $A = 0.5 \times 12 \times 14 \times 0.6428 = 53.99$ (not 63.99) So statement 10 is False. 5. Summary of answers: - 1: True - 2: False - 3: True - 4: False - 5: True - 6: True - 7: False - 8: True - 9: True - 10: False 6. Explanation in simplest terms: - Cosine is adjacent over hypotenuse because it measures how much the side next to the angle compares to the longest side. - Sine is opposite over hypotenuse because it measures how much the side across from the angle compares to the longest side. - Tangent is opposite over adjacent because it compares the side across the angle to the side next to it. - The hypotenuse is found by the Pythagorean theorem, which says the square of the longest side equals the sum of squares of the other two. - The Law of Cosines formula is for finding a side, not area. - The area formula with sine is for when you know two sides and the angle between them. - Heron's formula calculates area from all three sides. - Calculations must be precise; small errors change true/false answers.