Mensuration and trigonometry Revision Notes

Pythagoras in 3D Shapes

Pythagoras' theorem can be used to find lengths within three-dimensional shapes like cuboids, pyramids, or prisms. This often involves applying the theorem more than once. For instance, to find the space diagonal of a cuboid with dimensions length \(l\), width \(w\), and height \(h\):

1. First, find the diagonal of the base (or any face) using Pythagoras' theorem: \(d^2 = l^2 + w^2\).
2. Then, use this diagonal \(d\) and the height \(h\) to find the space diagonal \(D\): \(D^2 = d^2 + h^2\). Substituting the first result gives \(D^2 = l^2 + w^2 + h^2\). This extends the theorem to three dimensions.

Algebraic Applications of Pythagoras

In higher-level problems, the side lengths of a right-angled triangle may be given in terms of algebraic expressions. For example, if the sides are \(x\), \(x+1\), and \(x+2\) (where \(x+2\) is the hypotenuse), applying Pythagoras' theorem leads to an equation: \(x^2 + (x+1)^2 = (x+2)^2\). Expanding and simplifying this equation will result in a quadratic equation that needs to be solved for \(x\). Remember that lengths must be positive, so any negative solutions for \(x\) should be discarded. These problems often test your ability to set up and solve quadratic equations as well as apply the theorem correctly.

Pythagoras and Coordinate Geometry

Pythagoras' theorem is inherently linked to calculating the distance between two points in a coordinate plane. Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the horizontal distance between them is \(|x_2 - x_1|\) and the vertical distance is \(|y_2 - y_1|\). These form the two legs of a right-angled triangle whose hypotenuse is the distance between the two points. The distance \(d\) is found using the formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This is essentially a direct application of Pythagoras' theorem, where the change in x-coordinates and change in y-coordinates are the perpendicular sides.

Converse of Pythagoras' Theorem

The converse of Pythagoras' theorem is also important: if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle. This allows us to prove whether a given triangle is right-angled without direct observation or measurement of angles. If \(c^2 = a^2 + b^2\) for a triangle with sides \(a, b, c\) (where \(c\) is the longest side), then the angle opposite side \(c\) is \(90^\circ\). This is useful in proofs and problem-solving where the nature of an angle needs to be determined.

Labeling Sides of a Right-Angled Triangle

Before applying trigonometry, you must correctly label the sides of the right-angled triangle relative to the acute angle you are interested in. The 'hypotenuse' is always the longest side, opposite the right angle. The 'opposite' side is the one directly across from the chosen angle. The 'adjacent' side is next to the chosen angle, but it's not the hypotenuse. Carefully identify these three sides for every problem to avoid errors.

The Trigonometric Ratios: SOH CAH TOA

The three fundamental trigonometric ratios are Sine, Cosine, and Tangent. They define the relationship between an acute angle and the ratios of two sides in a right-angled triangle. Remember the acronym SOH CAH TOA:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Choose the correct ratio based on which angle and sides (known or unknown) are involved in your problem. For example, if you know the opposite side and the hypotenuse, and want to find the angle, you'd use Sine.

Finding Unknown Side Lengths

To find an unknown side length when you know an angle and one side:

1. Label the sides (Opposite, Adjacent, Hypotenuse) relative to the known angle.
2. Choose the correct trigonometric ratio (SOH CAH TOA) that involves the known side and the unknown side.
3. Set up the equation. For example, if you know the angle ($\theta$), the hypotenuse (H), and want to find the opposite side (O), you'd use $\sin(\theta) = \frac{O}{H}$, which rearranges to $O = H \times \sin(\theta)$.
4. Calculate the value using your calculator. Remember to check your calculator is in 'degree' mode.

Finding Unknown Angles

To find an unknown angle when you know two side lengths:

1. Label the sides (Opposite, Adjacent, Hypotenuse) relative to the unknown angle.
2. Choose the correct trigonometric ratio (SOH CAH TOA) that involves the two known sides.
3. Set up the equation. For example, if you know the opposite side (O) and the hypotenuse (H), you'd use $\sin(\theta) = \frac{O}{H}$.
4. Use the inverse trigonometric function (e.g., $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) to find the angle. For the example $\sin(\theta) = \frac{O}{H}$, you would calculate $\theta = \sin^{-1}\left(\frac{O}{H}\right)$. Your calculator has these inverse functions, usually accessed by pressing 'shift' or '2ndF' then the trig button. Ensure your calculator is in 'degree' mode.

Recalling SOH CAH TOA

The fundamental relationships in right-angled trigonometry are defined as SOH CAH TOA. SOH stands for Sine = Opposite / Hypotenuse. CAH stands for Cosine = Adjacent / Hypotenuse. TOA stands for Tangent = Opposite / Adjacent. Ensure you can correctly identify the opposite, adjacent, and hypotenuse sides relative to the angle you are working with. The hypotenuse is always opposite the right angle, the opposite side is opposite the specific angle you are considering, and the adjacent side is next to that specific angle (but not the hypotenuse).

Calculating Side Lengths

To find an unknown side length, first identify the known angle and the known and unknown sides relative to it. Then, select the appropriate sin, cos, or tan ratio. Substitute the known values and rearrange the formula to solve for the unknown side. For example, if you know an angle θ and the hypotenuse H, and need to find the opposite side O, use sin(θ) = O/H, so O = H * sin(θ). If you need to find H, use H = O / sin(θ).

Calculating Angles

To find an unknown angle, identify the two known side lengths relative to the angle you want to find. Select the correct sin, cos, or tan ratio using SOH CAH TOA. Substitute the known side lengths into the ratio. To find the angle, use the inverse trigonometric functions: arcsin (sin⁻¹), arccos (cos⁻¹), or arctan (tan⁻¹) on your calculator. For example, if you know the opposite side O and the adjacent side A, use tan(θ) = O/A, then θ = tan⁻¹(O/A).

Problem-Solving Strategies

Many problems require multiple steps or involve combining trigonometry with other geometric facts (e.g., Pythagoras' theorem, area formulas, properties of 3D shapes). Break down complex problems into smaller, manageable parts. Draw clear diagrams, label knowns and unknowns, and carefully check your calculator's mode (degrees or radians). Round your final answers to an appropriate degree of accuracy, typically 3 significant figures unless otherwise specified.

Exact Values for 30° and 60°

To derive the exact trigonometric ratios for 30° and 60°, start with an equilateral triangle of side length 2. Bisect one of the angles and the opposite side to form two right-angled triangles. Each right-angled triangle will have angles 30°, 60°, and 90°. The hypotenuse will be 2, one leg will be 1 (half of the original side), and the other leg will be $\sqrt{2^2 - 1^2} = \sqrt{3}$ by Pythagoras' theorem.

From this triangle: For 30°: sin(30°) = Opposite/Hypotenuse = 1/2 cos(30°) = Adjacent/Hypotenuse = $\sqrt{3}/2$ tan(30°) = Opposite/Adjacent = 1/$\sqrt{3}$ = $\sqrt{3}/3$

For 60°: sin(60°) = Opposite/Hypotenuse = $\sqrt{3}/2$ cos(60°) = Adjacent/Hypotenuse = 1/2 tan(60°) = Opposite/Adjacent = $\sqrt{3}/1 = \sqrt{3}$

Exact Values for 45°

To derive the exact trigonometric ratios for 45°, consider an isosceles right-angled triangle. Let the two equal sides be length 1. By Pythagoras' theorem, the hypotenuse will be $\sqrt{1^2 + 1^2} = \sqrt{2}$. The angles in this triangle are 45°, 45°, and 90°.

From this triangle: sin(45°) = Opposite/Hypotenuse = 1/$\sqrt{2}$ = $\sqrt{2}/2$ cos(45°) = Adjacent/Hypotenuse = 1/$\sqrt{2}$ = $\sqrt{2}/2$ tan(45°) = Opposite/Adjacent = 1/1 = 1

Exact Values for 0° and 90°

While not derived from simple triangles in the same way, the exact trigonometric ratios for 0° and 90° are essential. Imagine a point moving along a unit circle. As the angle approaches 0°, the y-coordinate approaches 0 and the x-coordinate approaches 1. As the angle approaches 90°, the y-coordinate approaches 1 and the x-coordinate approaches 0.

For 0°: sin(0°) = 0 cos(0°) = 1 tan(0°) = 0

For 90°: sin(90°) = 1 cos(90°) = 0 tan(90°) is undefined (division by zero).

Summary Table of Exact Values

It's helpful to remember these values, often by constructing the triangles or using a 'hand trick' for 0, 30, 45, 60, 90.

| Angle ($\theta$) | sin($\theta$) | cos($\theta$) | tan($\theta$) | |---|---|---|---| | 0° | 0 | 1 | 0 | | 30° | 1/2 | $\sqrt{3}/2$ | $\sqrt{3}/3$ | | 45° | $\sqrt{2}/2$ | $\sqrt{2}/2$ | 1 | | 60° | $\sqrt{3}/2$ | 1/2 | $\sqrt{3}$ | | 90° | 1 | 0 | Undefined |