Introduction to Trigonometry and Pythagoras' Theorem
Trigonometry Unlocked: From Side-Quest to Main Quest 📐✨
Introduction
1. Introduction
Yo! Welcome to the Trigonometry level-up guide. I know what you're thinking... 'Trig-o-what-now?' It sounds super complicated, but I promise it's not. Think of it as a secret toolkit for solving puzzles with triangles. It's used in everything from designing video games and creating special effects in movies to planning a flight path. By the end of this, you'll be a triangle master. Let's get this bread. 🍞
2. Pythagoras' Theorem: The Relationship Between Sides
First up, let's talk about the legend himself: Pythagoras. His theorem is your go-to move when you have a right-angled triangle and you know two side lengths, but need the third. The key is to identify the hypotenuse: that's the longest side, always opposite the right angle. The rule is simple: the square of the two shorter sides ( and ) add up to the square of the hypotenuse (). The formula is your new best friend: . Easy, right? It's like a 2-for-1 deal on sides.
It doesn't matter which shorter side you call and which you call . What matters is spotting : the hypotenuse is the one that doesn't touch the little square in the corner.
To find the hypotenuse, square the two shorter sides, add them, then square root. Think of a spot on a game map: the horizontal and vertical distances are and , and the straight-line shortcut is .
Worked example
Worked Example: Finding the Hypotenuse
Worked Example: Finding the Longest Side (Hypotenuse)
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the length of the hypotenuse.
- 1First, write down Pythagoras' theorem. We have the two shorter sides, and . We need to find .
- 2Sub in the values you know. Square them up.
- 3Now, to get by itself, we need to do the opposite of squaring, which is finding the square root. Get your calculator ready!
Answer
Worked example
Worked Example: Calculating a Missing Hypotenuse
That New TV Screen Size Mystery 📺
You want to buy a new TV, and the box says it's 120 cm wide and 68 cm tall. TV sizes are measured by their diagonal length. What is the actual screen size (the diagonal) to the nearest whole number?
- 1First, let's ID what we're working with. The width and height of the TV form the two shorter sides of a right-angled triangle ('a' and 'b'). The diagonal screen size is the hypotenuse ('c'), which is what we need to find.
- 2Now, let's write down the main formula, our trusty Pythagoras' Theorem.
- 3Time to sub in the numbers we know. We'll plug in the values for the width and height.
- 4Let's get squaring! Calculate the squares of 120 and 68, then add them together to find the value of .
- 5We have , but we need just 'c'. To undo the square, we do the opposite: find the square root. Get your calculator ready!
- 6Finally, calculate the square root and round it to the nearest whole number as the question asks. This is the TV's official screen size!
Answer
3. Finding a Shorter Side Using Pythagoras' Theorem
Okay, but what if you already have the hypotenuse (the 'final boss' side) and one of the shorter sides? It's like knowing the final score but missing one team's points. You just have to rearrange the formula. Instead of adding, you subtract the square of the short side you know from the square of the hypotenuse. So, it becomes or . Just remember: always start with the longest side when you subtract.
Worked example
Worked Example: Calculating a Shorter Side
Worked Example: Applying Pythagoras' theorem to find a shorter side
You're at a skate park. A ramp is 13 meters long (that's the hypotenuse) and it covers a horizontal distance of 12 meters along the ground. How high is the ramp?
- 1Identify your sides. The ramp itself is the hypotenuse, . The ground distance is a shorter side, . We need to find the height, . We'll use the rearranged formula.
- 2Plug in the numbers and calculate the squares.
- 3Find the square root to get the final answer. The ramp is 5 meters high! 🛹
Answer
4. The Trigonometric Ratios: SOH CAH TOA
Pythagoras is great, but he ghosts you when angles get involved. That's where Trigonometry steps in. For any right-angled triangle, we have three magic ratios. To use them, you need to stand at one of the non-right angles and label the sides from that perspective: Opposite (the side across from you), Adjacent (the side next to you, but not the hypotenuse), and the Hypotenuse (always the longest one).
The magic mnemonic to remember the ratios is SOH CAH TOA:
SOH: Sine() = Opposite / Hypotenuse
CAH: Cosine() = Adjacent / Hypotenuse
TOA: Tangent() = Opposite / Adjacent
The labels depend on which angle you stand at: switch to the other acute angle and Opposite and Adjacent swap, while the Hypotenuse stays put.
SOH: Sine() = Opposite / Hypotenuse
CAH: Cosine() = Adjacent / Hypotenuse
TOA: Tangent() = Opposite / Adjacent
The labels depend on which angle you stand at: switch to the other acute angle and Opposite and Adjacent swap, while the Hypotenuse stays put.
Pick the ratio that connects what you know with what you want. Keep your calculator in Degrees (DEG) mode, and give angles correct to one decimal place.
Beyond Core: naming these as 'angles of elevation and depression' is Extended content (E6.2.4). On a Core paper the angle is described in words instead, such as 'the sun's rays make an angle of with the ground'.
Beyond Core: naming these as 'angles of elevation and depression' is Extended content (E6.2.4). On a Core paper the angle is described in words instead, such as 'the sun's rays make an angle of with the ground'.
Worked example
Worked Example: Identifying Sides and Selecting a Ratio
Worked Example: Identifying Sides with SOH CAH TOA
In a right-angled triangle, angle is . The side opposite angle is 10 cm, and the hypotenuse is 17.4 cm. Which trig ratio would you use to connect these three pieces of information?
- 1Look at what you've got. You have an angle (), the side Opposite it (10 cm), and the Hypotenuse (17.4 cm).Given: Angle, Opposite, Hypotenuse
- 2Now check the mnemonic SOH CAH TOA. Which part uses O and H?SOH stands for Sine = Opposite / Hypotenuse. This is the one!
Answer
SOH stands for Sine = Opposite / Hypotenuse. This is the one!
5. Calculating a Missing Side Using the Sine Ratio
Let's put SOH to work. If you know an angle and the hypotenuse, you can find the opposite side. Or if you know the angle and the opposite side, you can find the hypotenuse. It's all about setting up the equation and solving for the unknown. Think of it like a formula in a game's crafting system: put in two ingredients (angle and one side) to get a new item (the missing side).
Worked example
Worked Example: Applying the Sine Ratio
Worked Example: Using the sine ratio to find a side length
A kite is flying on a 50m string. The string makes an angle of with the ground. How high is the kite above the ground? (Assume the string is a straight line).
- 1Draw a quick sketch! The string is the Hypotenuse (50m). The angle with the ground is . The height of the kite is the side Opposite the angle. We have O, H, and an angle, so we're using SOH.
- 2Plug in the values we know. Let the height be .
- 3To get on its own, we multiply both sides by 50. Type into your calculator. Make sure it's in Degrees (DEG) mode!
Answer
6. Using Cosine and Tangent to Find Missing Sides
SOH CAH TOA isn't just about Sine. You can use Cosine (CAH) or Tangent (TOA) in exactly the same way. The trick is to correctly identify which sides you have (Opposite, Adjacent, Hypotenuse) relative to your angle. Got the Adjacent and Hypotenuse? Use Cosine. Have the Opposite and Adjacent? Use Tangent. You've got this! It's like choosing the right filter for your pic – you pick the one that works with what you've got.
Worked example
Worked Example: Applying Trigonometric Ratios for Sides
Worked Example: Using trigonometric ratios to find a side length
You're standing on level ground, 20 meters away from the base of a tall building. The straight line from your feet to the top of the building makes an angle of with the ground. How tall is the building?
- 1Let's label our triangle. The distance from you to the building is the Adjacent side (20m). The height of the building is the Opposite side (what we want to find). The angle is . We have O and A, so it's time for TOA.
- 2Sub in your values, with for the height (Opposite).
- 3Multiply both sides by 20 to solve for . Calculator time!
Answer
Worked example
Worked Example: Height of a Tree with Tangent
Worked Example: How Tall Is That Tree?
A point on level ground is 100m from the base of a very tall vertical tree. The straight line from that point to the top of the tree makes an angle of with the ground. What is the height of the tree?
- 1This is a classic TOA problem. The distance from the tree is the Adjacent side (100m). The height of the tree is the Opposite side. The angle with the ground is our .
- 2Plug in the numbers.
- 3Solve for the Height by multiplying.
Answer
Worked example
Worked Example: Calculating a Missing Side
Let's Build That Skate Ramp! 🛹
You're designing a new skate ramp. The ramp itself will be 3.5 meters long and will make an angle of with the ground. What is the horizontal distance the ramp will cover along the ground? Give your answer correct to one decimal place.
- 1First, let's visualize and label the triangle. The ramp is the Hypotenuse (the longest side). The horizontal distance along the ground is Adjacent to the angle. The height of the ramp would be the Opposite side, but we don't need that here. We know the angle and the hypotenuse, and we want to find the adjacent side.Knowns: Angle = , Hypotenuse = m
Unknown: Adjacent = - 2Now we need to pick our tool from SOHCAHTOA. We have the Adjacent (A) and the Hypotenuse (H). The ratio that connects A and H is CAH. So, we'll be using the cosine ratio.
- 3Let's substitute our values into the formula. The angle is , the Adjacent side is our unknown , and the Hypotenuse is m.
- 4Time to solve for . To get by itself, we need to undo the division by 3.5. We do this by multiplying both sides of the equation by 3.5.
- 5Finally, grab your calculator (make sure it's in DEG mode!). Type in the calculation and round the result to one decimal place as the question asks.
So, the ramp will cover a horizontal distance of 3.2 meters.
Answer
So, the ramp will cover a horizontal distance of 3.2 meters.
So, the ramp will cover a horizontal distance of 3.2 meters.
7. Finding an Angle Using Inverse Trigonometric Functions
So we can find sides. But what if you have the sides and need to find the angle? Like, you know your part-time job is a 5km walk East and a 2km walk North from home, but what's the angle of that direction? For this, we use the inverse trig functions: , , and . They look like they have an exponent of -1. These functions 'undo' the normal sin, cos, and tan to give you back the angle. If , then . It's like hitting 'undo' or 'rewind' to find out what the original angle was.
Worked example
Worked Example: Calculating an Unknown Angle
Worked Example: Using trigonometric ratios to find an angle
A rectangular phone screen has a width of 7 cm and a height of 15 cm. What angle does the diagonal make with the shorter side (the width)?
- 1Picture the rectangle. The diagonal cuts it into two right-angled triangles. From the corner, the height (15 cm) is the Opposite side and the width (7 cm) is the Adjacent side. We have O and A, so we'll use TOA again.
- 2Now we need to find the angle . We use the inverse tangent function, which is probably a SHIFT + TAN button on your calculator.
- 3Calculate the value and give it to one decimal place, as Core angle answers should be.
Answer
8. Solving Multi-Step Problems in Two Dimensions
This is where it all comes together. Some problems are multi-step, requiring you to use Pythagoras and SOH CAH TOA in the same question. It's like a boss battle with two phases. You might need to use Pythagoras to find a missing side first, and then use that side in a SOH CAH TOA calculation to find an angle, or vice-versa. Don't panic! Just break the problem down. Draw a diagram, label everything you know, and figure out what you need to find in step one. One step at a time. 🎯
Worked example
Worked Example: Solving a 2D Problem
Worked Example: Solving 2D problems using both Pythagoras and trigonometry
Two friends start at the same point O. Amir walks 90m due East and stops at A. Bella walks 60m due North and stops at P. Find the distance AP, and the bearing of A from P. (A bearing is an angle measured clockwise from North.)
- 1Sketch it: O, A and P make a right-angled triangle with the right angle at O. The distance AP is the hypotenuse, so use Pythagoras.
- 2Now the angle at P, between the line PO (pointing due South) and the line PA. From P, the side OA (90m) is Opposite and PO (60m) is Adjacent, so use TOA.
- 3Turn it into a bearing. Facing North at P and turning clockwise, due South is . The line to A is short of South (A is to the south-east of P), so subtract.
Answer
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