# Trigonometry: Right-angled triangle

• MHB
• Brian Bart
In summary, Trigonometry is a branch of mathematics that deals with the study of relationships between the sides and angles of triangles. A right-angled triangle is a triangle that has one angle measuring 90 degrees, represented by a square symbol. The three basic trigonometric ratios are sine, cosine, and tangent, which are used to relate the angles of a right-angled triangle to the lengths of its sides. To find the missing side length of a right-angled triangle, the Pythagorean theorem or one of the trigonometric ratios can be used. Trigonometry has various real-life applications, including architecture, engineering, navigation, astronomy, and everyday activities like measuring heights and distances.
Brian Bart

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part (i) of the student's response is correct.

part (ii) has an algebra error in determining the length of DC

if $DC^2 = DB^2 - CB^2$, then $DC \ne DB - CB$

the trig for part (iii) is correct ... can't say I agree with using 120 degrees for angle ADB since it induces rounding error in determining the length of AB.

Using the equation $\sin(40) = \dfrac{BC}{AB} \implies AB = \dfrac{BC}{\sin(40)}$ will not induce that error.

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## 1. What is a right-angled triangle?

A right-angled triangle is a triangle with one angle measuring 90 degrees. The other two angles are acute, meaning they are less than 90 degrees.

## 2. What is the Pythagorean theorem?

The Pythagorean theorem is a fundamental formula in trigonometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. It can be written as a² + b² = c², where c is the length of the hypotenuse and a and b are the lengths of the other two sides.

## 3. How do you find the missing side length in a right-angled triangle?

To find the missing side length in a right-angled triangle, you can use the Pythagorean theorem or one of the trigonometric ratios (sine, cosine, or tangent) depending on the given information. If you know the lengths of two sides, you can use the Pythagorean theorem to find the third side. If you know one side and one of the acute angles, you can use the trigonometric ratios to find the missing side length.

## 4. What are the trigonometric ratios?

The trigonometric ratios are sine, cosine, and tangent, which are used to relate the angles and sides of a right-angled triangle. Sine is the ratio of the length of the side opposite an angle to the length of the hypotenuse. Cosine is the ratio of the length of the adjacent side to the length of the hypotenuse. Tangent is the ratio of the length of the opposite side to the length of the adjacent side.

## 5. How is trigonometry used in real life?

Trigonometry is used in many real-life applications, such as architecture, engineering, navigation, and astronomy. It is used to calculate distances, heights, and angles in various structures and objects. For example, architects use trigonometry to design buildings and engineers use it to calculate the slope of roads and bridges. In navigation, trigonometry is used to determine the position of a ship or plane. In astronomy, it is used to calculate the distance between celestial objects.

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