From the statement, we have a right triangle with:
• angle θ,
,• opposite side OS = 7,
,• hypotenuse H = 12,
and we must find the value of angle θ.
We also have Tom's resolution of the problem.
(1) From the resolution, we see that Tom computes the angle θ using the following formula:
[tex]\cos\theta=\frac{7}{12}.[/tex]Adding the data above the sides, we have:
[tex]\cos\theta=\frac{7}{12}=\frac{OS}{H}\Rightarrow\cos\theta=\frac{OS}{H}\text{ }✖[/tex]From trigonometry, we know that this equation is wrong. The correct trigonometric relation is:
[tex]\sin\theta=\frac{OS}{H}.[/tex](2) Replacing the values OS = 7 and H = 12 in the correct trigonometric relation, we have:
[tex]\sin\theta=\frac{7}{12}.[/tex]Solving for θ, we get:
[tex]θ=\sin^{-1}(\frac{7}{12})\cong35.7\degree.[/tex]Answer• The mistake in Tom's resolution is that he used the incorrect trigonometric relation for the angle, the opposite side and the hypotenuse.
,• Using the correct trigonometric relation, which involves a sine function instead of the cosine, we get θ ≅ 35.7°.
