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Introduction to Trigonometry
CH08 Question Bank
CBSE_CH08_QUESTION_BANK
Grade 10
Question:
[Case Study]
A student is designing a company logo using two right triangles of different sizes but with the same shape (i.e. the same acute angle $\theta$ at one corner in both). In the smaller triangle, the side opposite $\theta$ is $3$ cm and the hypotenuse is $5$ cm. In the larger triangle, the hypotenuse is $15$ cm.
(a) Find $\sin\theta$ using the smaller triangle. [1 Mark] (b) Since both triangles have the same angle $\theta$, what can you say about the value of $\sin\theta$ computed from the larger triangle? [1 Mark] (c) Find the side opposite $\theta$ in the larger triangle. [1 Mark] (d) Find $\cos\theta$ for the smaller triangle. [1 Mark]
Step-by-Step Solution
Key Concept: Case study on trigonometry ratios and identities.
(a) Find $\sin\theta$ using the smaller triangle. [1 Mark] $\sin\theta=\dfrac{3}{5}$. [1.0 Mark]
(b) Since both triangles have the same angle $\theta$, what can you say about the value of $\sin\theta$ computed from the larger triangle? [1 Mark] Since the ratios of a given angle are always the same regardless of the triangle's size, $\sin\theta$ computed from the larger triangle will also equal $\dfrac35$. [1.0 Mark]
(c) Find the side opposite $\theta$ in the larger triangle. [1 Mark] $\sin\theta=\dfrac35=\dfrac{\text{opposite}}{15}\Rightarrow\text{opposite}=9$ cm. [1.0 Mark]
(d) Find $\cos\theta$ for the smaller triangle. [1 Mark] $\cos^2\theta=1-\sin^2\theta=1-\dfrac{9}{25}=\dfrac{16}{25}\Rightarrow\cos\theta=\dfrac45$. [1.0 Mark]
Correct Answer:$\sin\theta=\dfrac{3}{5}$. [1.0 Mark] | Since the ratios of a given angle are always the same regardless of the triangle's size, $\sin\theta$ computed from the larger triangle will also equal $\dfrac35$. [1.0 Mark] | $\sin\theta=\dfrac35=\dfrac{\text{opposite}}{15}\Rightarrow\text{opposite}=9$ cm. [1.0 Mark] | $\cos^2\theta=1-\sin^2\theta=1-\dfrac{9}{25}=\dfrac{16}{25}\Rightarrow\cos\theta=\dfrac45$. [1.0 Mark]
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