Basic Trigonometry and Right-Angled Triangle Exercises
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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. $\tan \theta = \frac{2}{\sqrt{5}}$
Animated Video Solution
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Step by Step Written Solution
Let's work through the first problem, which asks us to identify the incorrect statement about a right-angled triangle.
Problem 1: Right-Angled Triangles
Option A says the angle opposite to the hypotenuse is a right angle. This is true by definition. In any right triangle, the longest side is the hypotenuse and it sits opposite the ninety-degree angle.
Option B presents the Pythagorean theorem, which states that the sum of the squares of the base and height equals the square of the hypotenuse. This is a fundamental property of right triangles.
Option C describes an isosceles right-angled triangle. By definition, an isosceles triangle has two equal sides. So this statement is also correct.
Finally, look at Option D. It suggests that cosine of theta can be greater than one. However, in a right triangle, cosine is the adjacent side over the hypotenuse. Since the hypotenuse is always the longest side, this ratio can never exceed one.
Therefore, the incorrect statement is D.
Now, let's look at problem five. We are given that sine of theta equals two thirds for an acute angle theta.
Problem 5: Trigonometric Ratios
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