Use the sine, cosine and tangent ratios to solve problems involving right-angled triangles where angles are measured in degrees, or degrees and minutes
Solve problems involving the use of trigonometry in two and three dimensions
Interpret information about a two or three-dimensional context given in diagrammatic or written form and construct diagrams where required
T1.2 Radians
Define and use radian measure and understand its relationship with degree measure
Recognise and use the exact values of , and in both degrees and radians for integer multiples of and
There are three very useful ratios that can be used to relate the sides and angles of a right angled triangle. These ratios are shown below:
Side / Angle Relationship of Right Triangles
Relationship
Acronym
SOH
CAH
TOA
This is often referred to by its acronym SOH CAH TOA. A further explanation of this acronym is shown below:
Explanation:SOH CAH TOA Acronym
To understand this acronym we need must first understand the basic definitions shown below:
From this we can then form the ratios using the image shown below:
Examples that utilise these three ratios are shown below:
Missing Side Example
We can find the missing sides x and h using the SOH CAH TOA ratios.
Finding H
We can solve for H using the SOH ratio:
We can then plug in our numbers and solve for h
Finding X
We can solve for x using the TOA ratio:
We can then plug in our numbers and solve for x
Missing Angle Example
We can solve for θ using the CAH ratio
We can then plug in our numbers and solve for θ.
Pythagoras’ Theorem
Pythagoras’ theorem helps us find a missing side of a right angled triangle, given the other two sides. The theorem is stated below:
Pythagoras’ Theorem
An example of Pythagoras’ theorem can be seen below:
Pythagoras’ Theorem Example
If we want to find the missing side in this triangle we can apply Pythagorus’ theorem.
Substituting our values gives:
We can now rearrange and solve for .
Finding One Ratio Given Another
If we are given a trigonometric ratio, we can draw the triangle which corresponds to that ratio. For example:
Pythagoras’ theorem can then be used to find the missing side. Once the missing side is found we can find expressions for all of the other trigonometric ratios. An example of this can be seen below:
Finding One Trig Ratio Given Another
Given and θ is in the second quadrant, find the value of and
To start this we will draw the original triangle:
We can then find the missing side using Pythagoras’ theorem.
Finding Cos
We can now find the value of using the CAH ratio. However, we also have to note that the answer must be negative as we are dealing with the second quadrant:
Finding Tan
We can now find the value of using the TOA ratio. However, we also have to note hat the answer must be negative as tan is negative in the second quadrant:
Special Triangles
There are two special triangles which gives us exact valued solutions to the trigonometric ratios. These are known as the 45-45-90 and 30-60-90 triangles, which are shown below:
caption
From these two triangles, we can write the exact output of each trigonometric expression. This is summarised in the table below:
Lesson Loading
Difficulty
00
Time
SOLUTION
Difficulty
3 Mins
Time
Write down the values of , , , given the following diagram.
SOLUTION
We can find the hypotenuse using Pythagoras’ theorem:
Hence we can now express the missing expressions as:
Difficulty
2 Mins
Time
Find the missing side of the triangle shown below. Express your answer to 2 decimal places.
SOLUTION
We use the CAH ratio to form a relationship between the provided variables.
We can now solve for to give us:
Difficulty
3 Mins
Time
It is given that is the angle inside of a triangle and .
Draw a triangle that shows the given information
Use Pythagoras’ theorem to find the missing side of the triangle
Hence, find and assuming is in the first quadrant.
SOLUTION
Part 1
Tan represents the opposite side over the adjacent side. We can therefore draw this as:
Part 2
To find the hypotenuse we must apply Pythagoras’ theorem.
Part 3
We can express the values of sine and cosine using our right angled trig expressions. Note: sin and cos are both positive since we are told that θ is in the first quadrant.
Difficulty
3 Mins
Time
Show that the following equality holds without using a calculator.
SOLUTION
We can use the values from our special triangles to help us simplify this algebraically.
We can hence replace the trigonometric expressions with their exact values and simplify.
We can then rationalise the denominator to get:
If we check the value of the RHS we can see that it is also . Therefore, we have proven the expression as LHS = RHS.
Difficulty
2 Mins
Time
Given the following diagram, write down the values of , , and .
SOLUTION
To start this question we must first find the hypotenuse of the triangle using Pythagoras’ theorem.
From here we can use SOH CAH TOA to write down the values for our trigonometric ratios:
Difficulty
4 Mins
Time
If find all possible solutions for
SOLUTION
To start this question we will draw a triangle that contains all our given information:
We can then find the missing side of the triangle using Pythagoras’ theorem.
At this stage we can find the trigonometric ratio for sin using SOH. However, we must also note that tan was positive meaning θ could be in the first or third quadrant. Because of this sin is potentially positive from the first quadrant or negative from the third quadrant, hence:
Difficulty
4 Mins
Time
Find the exact value of the expression below:
Leave your answer in simplest form.
SOLUTION
We can use the special triangles shown below to simplify this expression
Subbing in our values gives us:
We can then join the fractions in the numerator by creating a common denominator:
We can then simplify this nested fraction by noting that
We can then rationalise the denominator:
Further simplifying gives:
Difficulty
3 Mins
Time
If and , find the value of .
SOLUTION
We begin this question by drawing a right angle triangle which contains the information from the given trig ratio:
From this we can find the missing side using Pythagoras’ Theorem
To find the value of we must first understand which quadrant θ came from. Since is positive and is negative, θ must be from the fourth quadrant. In this region we know that is negative, hence:
Difficulty
4 Mins
Time
Use right angled trigonometry to find the value of θ in the following triangle
Express your answer in degrees ( 2dp )
SOLUTION
To find the value of θ we will draw a line straight down the centre, forming a right angle with the base:
We can now find the value of θ using the CAH ratio
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Presets
Trigonometry with Right-Angled Triangles
5A - Cambridge Advanced Year 11
Question 1 (a, b, e)
Question 3 (a, f)
Question 4 (d, h)
Question 9 (a, b{i, ii})
Question 10 (a, b, c)
Question 11 (b, d)
Question 13 (b, d, f)
Question 15 (all)
Question 18 (b, c, d)
You want to substitute the special ratios and simplify, reciprocals are your friends here
Quadrant, Sign, and Related Acute Angle
5E - Cambridge Advanced Year 11
Question 5 (a, c, g, j, l)
You’ll want to use the unit circle identities like sin(theta) = sin(180-theta)
Question 6 (a, f, g, i)
Question 10 (a, d, h, l)
Question 12 (a, d, f)
Given one Trigonometric Function, Find Another
5F - Cambridge Advanced Year 11
Question 1 (a, b, d)
Question 3 (b)
Question 4 (b)
Question 5 (b, d)
Question 7 (a, b, d)
Question 11
Try drawing a triangle and using pythagorus’ theorem
Question 13 (all)
Lesson Loading
×
×
Right Angle Ratios
Review Questions
We extend the trigonometric ratios to right angled triangles by:
Multiplying the radius of the unit circle by H
Converting the unit circle to the unit triangle
Dividing the trigonometric ratios by each other
The three sides of a triangle are labeled with an O, A, H. What do these letters represent?
The sides Opposite and Adjacent to our angle, and and the Hypotenuse
The side Over and Above our angle, and the Hypotenuse
The side Over and Around our longest side, and the Head of the triangle
What can we do with the SOH, CAH and TOA ratios?
Solve for missing angles or sides on any triangle
Solve for missing angles and sides on a right angle triangle
Solve for missing sides, but not angles on a right triangle
Solve for missing angles, but not sides on any triangle
×
Right Angle Ratios
Review Questions
We extend the trigonometric ratios to right angled triangles by:
Multiplying the radius of the unit circle by H
Dividing the trigonometric ratios by each other
Converting the unit circle to the unit triangle
The three sides of a triangle are labeled with an O, A, H. What do these letters represent?
The side Over and Around our longest side, and the Head of the triangle
The side Over and Above our angle, and the Hypotenuse
The sides Opposite and Adjacent to our angle, and and the Hypotenuse
What can we do with the SOH, CAH and TOA ratios?
Solve for missing angles, but not sides on any triangle
Solve for missing sides, but not angles on a right triangle
Solve for missing angles and sides on a right angle triangle
