Use trigonometry to work out the value of $x$ in the diagram below. Give your answer correct to two decimal places. The triangle has one angle measuring $20^{ ext{°}}$ and the opposite side measuring $6$. A different right-angled triangle is shown on the right. It has sides of length $a$, $b$, and $c$. One of the angles is $Y$. In this triangle, $a + b > c$. Use this to prove that: $$ ext{cos } Y + ext{sin } Y > 1$$

Junior Cycle Mathematics Trigonometry: Use trigonometry to work out the

Use trigonometry to work out the value of $x$ in the diagram below - Junior Cycle Mathematics - Question 13 - 2019

Question 13

Use trigonometry to work out the value of $x$ in the diagram below. Give your answer correct to two decimal places. The triangle has one angle measuring $20^{ ext{... show full transcript

Worked Solution & Example Answer:Use trigonometry to work out the value of $x$ in the diagram below - Junior Cycle Mathematics - Question 13 - 2019

Step 1

Use trigonometry to work out the value of x in the diagram below.

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Answer

To find the value of $x$ , we can use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, we have:

$an(20^{ ext{°}}) = \frac{6}{x}$

Rearranging the equation gives:

$x = \frac{6}{\tan(20^{ ext{°}})}$

Using a calculator, we find:

$x \approx \frac{6}{0.3640} \approx 16.48$

Therefore, the value of $x$ is approximately $16.48$ .

Step 2

Use this to prove that: cos Y + sin Y > 1

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Answer

In the triangle with sides $a$ , $b$ , and $c$ , we are given that:

$a + b > c$

We can relate this to the trigonometric functions:

By definition, we have:

$\cos Y = \frac{b}{c} \quad \text{ and } \quad \sin Y = \frac{a}{c}$

Now, substituting these values into the inequality:

$\cos Y + \sin Y = \frac{b}{c} + \frac{a}{c} = \frac{a + b}{c}$

Since $a + b > c$ , we divide both sides by $c$ (noting that $c > 0$ ):

$\frac{a + b}{c} > 1$

Thus, we arrive at:

$\cos Y + \sin Y > 1$

This proves the statement.

Use trigonometry to work out the value of $x$ in the diagram below - Junior Cycle Mathematics - Question 13 - 2019

Worked Solution & Example Answer:Use trigonometry to work out the value of $x$ in the diagram below - Junior Cycle Mathematics - Question 13 - 2019

Use trigonometry to work out the value of x in the diagram below.

Use this to prove that: cos Y + sin Y > 1

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