3.9.2 Right-angled Trigonometry
3.9.2 Right-angled Trigonometry
3.9.2 Right-angled Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
In particular, it studies the functions and properties of the three basic trigonometric ratios: sine, cosine, and tangent, and their reciprocals: cosecant, secant, and cotangent.
The basic trigonometric ratios are defined as follows:
Sine \( (sin) \): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
\( sinθ=\frac{opposite}{hypotenuse}=\frac{O}{H} \)
Cosine \( (cos) \): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
\( cosθ=\frac{adjacent}{hypotenuse}=\frac{A}{H} \)
Tangent \( (tan) \): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
\( tanθ=\frac{opposite}{adjacent}=\frac{O}{A} \)
\( SOH \) – \( CAH \) – \( TOA \) is an easy way to remember the trigonometry ratios.
The reciprocals of these functions are defined as follows:
Cosecant \( (csc) \): The cosecant of an angle is the reciprocal of the sine function:
\( csc θ=\frac{1}{sinθ} \)
Secant \( (sec) \): The secant of an angle is the reciprocal of the cosine function:
\( sec θ=\frac{1}{cosθ} \)
Cotangent \( (cot) \): The cotangent of an angle is the reciprocal of the tangent function:
\( cot θ=\frac{1}{tanθ} \)
Trigonometry is used to solve many problems involving angles and distances.
For example, it can be used to find the height of a building, the distance between two points, or the angle of elevation of an object.
Trigonometry is also used extensively in calculus, which is another branch of mathematics that deals with rates of change and integration.
Finding missing lengths:
Trigonometric ratios can be used to find missing lengths in a right triangle.
There are three main trigonometric ratios: \( sine \), \( cosine \), and \( tangent \).
Each of these ratios relates the length of two sides of a right triangle to one of its angles.
To use trigonometric ratios to find a missing length, follow these steps:
Identify which angle you are working with and which side length is missing.
Label the angle with a Greek letter (such as theta, θ ) and label the missing side with a lower-case letter (such as \( x \) ).
Determine which trigonometric ratio relates the angle and the side length you are working with.
Remember, sine \( (sin) \), cosine \( (cos) \), and tangent \( (tan) \) are the three main trigonometric ratios.
Write out the trigonometric ratio that relates the angle and the side length you are working with.
For example, if you are working with the angle theta and the opposite side length \( x \), you would use the sine ratio:
\( sin(θ)=\frac{opposite}{hypotenuse} \)
Substitute in the known values for the other side lengths and angles.
For example, if you know the adjacent side length is \( 6 \) and the angle theta is \( 30 \) degrees, you would have:
\( cos(30)=\frac{6}{x} \)
Solve for the missing side length.
\( x =\frac{6}{cos(30)} \)
\( x=4 \sqrt{3} \)
Finding missing angles:
Trigonometric ratios are used to relate the angles and sides of a right triangle.
The three primary trigonometric ratios are \( sine \), \( cosine \), and \( tangent \).
These ratios are commonly abbreviated as \( sin \), \( cos \), and \( tan \), respectively.
To use trigonometric ratios to find missing angles in a right triangle, you will need to follow these steps:
Identify the right triangle
Identify the right triangle, which is a triangle that has one angle that measures \( 90 \) degrees (a right angle).
Identify the sides and the angle you know
Label the sides of the triangle as the hypotenuse, adjacent, and opposite.
Identify the angle you know or given.
Choose the trigonometric ratio
Choose the appropriate trigonometric ratio to find the unknown angle based on the information you have.
Solve for the unknown angle
Using the ratio and the information you have, solve for the missing angle.
Here are the three primary trigonometric ratios and how they are used:
\( Sine (sin) =\frac{Opposite}{Hypotenuse} \) , \( Cosine (cos) = \frac{Adjacent}{Hypotenuse} \) ,\( Tangent (tan)=\frac{Opposite}{Adjacent} \)
Worked example:
In the diagram \( AB \) and \( CD \) are parallel.
\( AD \) and \( BC \) intersect at right angle at the point \( X \).
\( AB=10 \) cm, \( CD=5 \) cm, \( AX=8 \) cm and \( BX=6 \) cm.
Calculate angle \( XAB \).
As \( XAB \) is a right-angled triangle and all the three lengths are given, so we can
use any of the three trigonometric ratios.
Let’s use tangent ratio here, which is
\( tan(θ) = \frac{Opposite}{Adjacent} \)
\( tan (XAB) = \frac{6}{8} \)
\( XAB = \frac{6}{8} \)
\( XAB=36.86989765… \)
Correcting to one decimal place
\( XAB=36.9 \)
\( XAB=36.9° \)
Elevation and Depression
Elevation and depression are two terms used in trigonometry to describe the angles that are above or below the horizontal plane.
Elevation is the angle between the horizontal plane and an object or point above the plane.
It is measured as a positive angle from the horizontal plane.
For example, if you are standing on the ground and looking up at a bird in the sky, the angle of elevation is the angle between the ground and your line of sight to the bird.
Depression is the angle between the horizontal plane and an object or point below the plane.
It is measured as a negative angle from the horizontal plane.
For example, if you are standing on a bridge and looking down at a river below, the angle of depression is the angle between the horizontal plane and your line of sight to the river.
Both angles of elevation and depression can be used to calculate the height or depth of an object or point, given the distance from the observer and the angle.
These calculations are often used in fields such as engineering, surveying, and navigation.
The trigonometric functions used to calculate the height or depth of an object or point are the same as those used for finding missing angles in a right triangle.
The primary trigonometric functions are sine, cosine, and tangent.
The specific function used depends on the given information and what is being solved for.
Test yourself
Question 1:
\( ABDF \) is a parallelogram and \( BCDE \) is a straight line.
\( AF=12 \)cm, \( AB=9 \) cm, angle \( CFD=40° \)and angle \( FDE=80° \).
Calculate the height, \( h \), of the parallelogram.
[2]
Question 2:
The diagram shows a rectangle and a diagonal of the rectangle.
Work out the length of the diagonal of the rectangle.
Give your answer correct to one decimal place.
[3]
Question 3:
Here is part of a field.
This part of the field is in the shape of trapezium.
A farmer wants to put a fence all the way around the edge of this part of the field.
The farmer has \( 50 \) m of fence.
Does he have enough fence?
You must show all your working.
[5]
Question 4:
A solid metal cone has radius \( 1.65 \)cm and slant height \( 4.70 \)cm.
Find the angle that slant height makes with the base of the cone.
[2]
Question 5:
The diagram shows a field \( ABCD \).
The bearing of \( B \) from \( A \) is \( 140° \).
\( C \) is due east of \( B \) and \( D \) is due north of \( C \).
\( AB=400 \) m, \( BC=350 \)m and \( CD=450 \)m.
Calculate the distance from \( D \) to \( A \).
[6]