Exercises on 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 $\Delta ABC$, 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
Let's work through problem number two together. We are given a triangle A B C, which is right-angled at B. We know side A B is twenty four centimeters and B C is seven centimeters.
Problem 2: Right Triangle Trigonometry
To find the trigonometric ratios, we first need to determine the length of the hypotenuse, A C, using the Pythagorean theorem.
Plugging in our values, we have twenty four squared plus seven squared equals A C squared.
Calculating the squares gives us five hundred seventy six and forty nine. Adding them up, we get six hundred twenty five.
Taking the square root, we find that A C equals twenty five centimeters.
Now let's find the ratios for part a, centered around angle A. Sine is opposite over hypotenuse, which is B C over A C.
Part a: Ratios for Angle A
Substituting the lengths, sine of A is seven over twenty five.
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