Properties of Right-Angled Triangles and Trigonometric Ratios
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1. Which one of the following is not correct about a right-angled triangle?
A. The angle opposite to the hypotenuse is a right angle
B. If the base, height and hypotenuse of a right-angled triangle has lengths $b, p$ and $h$ units, respectively, then $b^2 + p^2 = h^2$.
C. For an isosceles right-angled triangle, two sides of a triangle are equal in length.
D. If $\theta$ is one of the angles of a right-angled triangle, then $\cos \theta$ can be greater than 1.
2. In $\triangle ABC$, right-angled at $B$, $AB = 24 \text{ cm}$, $BC = 7 \text{ cm}$. Determine
a. $\sin A$, $\cos A$
b. $\sin C$, $\cos C$
3. State whether the following are true or false. Justify your answer.
a. The value of $\tan A$ is always less than 1.
b. $\sin \theta = \frac{4}{3}$, for some acute angle $\theta$
c. If $\sin \theta = \frac{1}{3}$, then $\cos \theta = \frac{2\sqrt{2}}{3}$.
d. When $0^\circ \le \theta \le 90^\circ$ is an angle of a right-angled triangle, both $\sin \theta$ and $\cos \theta$ are between 0 and 1.
4. Given $0^\circ < \theta < 90^\circ$. If $\tan \theta = 1$, then which one of the following is not true?
A. $\cos \theta = \frac{\sqrt{2}}{2}$
B. $\cos \theta = \sin \theta$
C. $\theta = 45^\circ$
D. $\tan \theta = \sin \theta$
5. If $\sin \theta = \frac{2}{3}$ for an acute angle $\theta$, then which one of the following is correct?
A. $\cos \theta = \frac{\sqrt{5}}{2}$
B. $\tan \theta = \frac{2}{\sqrt{5}}$
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Step by Step Written Solution
In this problem, we need to find which statement about right-angled triangles is not correct. Let's evaluate each option one by one.
Right-Angled Triangles Analysis
Option A states that the angle opposite to the hypotenuse is a right angle. By definition, the hypotenuse is the side opposite the ninety-degree angle, so this statement is true.
A. Opposite to hypotenuse = 90° (True)
Option B presents the Pythagorean theorem. If the legs are b and p, and the hypotenuse is h, then b squared plus p squared equals h squared. This is a fundamental property of right triangles, so it is also true.
B. Pythagorean Theorem (True)
Option C mentions an isosceles right-angled triangle. By definition, an isosceles triangle has at least two equal sides. In this case, the two legs must be equal. So, this is also true.
C. Isosceles = two sides equal (True)
Now let's look at Option D. It says that the cosine of an angle in a right triangle can be greater than one.
Evaluating Trigonometric Limits
D. If $\theta$ is an angle of a right-angled triangle, then $\cos \theta$ can be greater than 1.
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