GM5-10: Apply trigonometric ratios and Pythagoras' theorem in two dimensions.
This means that students will apply trigonometric ratios to find the angles and lengths of sides in right-angled triangles. Students need to recognise two features of trigonometric ratios:
- Given similar right angled triangles the ratios of side lengths are the same, for example
For both triangles the ratio of the sides opposite and adjacent to angle A is 6/8 = 0.75. For any similar triangle this is also true. This ratio is the tangent of angle A, so A = 37°.
- The trigonometric ratios can be found using a right-angled triangle with a hypotenuse of one and applied to any other similar right angled triangle by scaling.
The trigonometric ratios are:
- sin θ = side opposite θ/hypotenuse,
- cos θ = side adjacent to θ/hypotenuse,
- tan θ = side opposite θ/side adjacent to θ
These ratios are often remembered using the mnemonic SOH CAH TOA.
Students will use Pythagoras’ theorem (a2 + b2 = c2) to find the lengths of sides of right angle triangles.
- measure lengths and angles accurately
- find the height of objects using trigonometry
- Showing that a polygon is composed of rectangles and triangles.
- Showing that a non-right angled triangle is composed of two right-angled triangles.
- Given the length of one side, finding the area of a regular polygon.
- Know some of the history behind Pythagoras’ theorem.
- Apply Pythagoras theorem in mathematical and real world contexts.
- Discover what sin, cos and tan are.
- Apply trigonometry ratios in mathematical and real world contexts.
- find lengths of objects using Pythagoras’ Theorem
- understand how similar triangles can be used to prove Pythagoras’ Theorem
- understand that Pythagoras’ Theorem can be thought of in terms of areas on the sides of the triangle
- Measure the lengths of the sides of sets of similar right angled triangles and find the ratio of sides.
- Investigate the relationship between these ratios and the angle size.
- Use calculators or tables to find the sine, cosine and tangent of angles.
- Apply the known ratios of unit triangles300
- Use cos to solve problems involving right-angled triangles
- Solve equations of the form cos(θ) = a, for a between –180 and 360 degrees
- State the value of cos(θ) in special cases
- Graph y = cos(θ)
- use sin to solve problems involving right-angled triangles
- solve equations of the form sin(θ) = a, for θ between –180º and 360º
- state the value of sin(θ) in special cases
- graph y = sin(θ)
- describe some of the ways in which the sine, cosine and tangent functions are related
- use tan to solve problems involving right-angled triangles
- solve equations of the form tan(θ) = a, for a between –180º and 360º degrees
- state the value of tan(θ) in special cases
- graph y = tan(θ)
solve problems using trigonometry
solve problesm using Pythagoras
explore Pythagoras' theorem
- state and explain Pythagoras’ theorem
- use Pythagoras’ theorem to find unknown sides of right angled triangles
Students develop their skills and knowledge on the mathematics learning progressions measurement sense, using maps and measuring tapes and/or supplied measurements to find, describe and use the steepness of a street. Students will be able to describe the steepness of a street in terms of a gradient300
- describe and demonstrate how trigonometry can be used to find the height of a tall building or tree
- describe and demonstrate how trigonometry can be used to find the height of a high hill, or other high object where one cannot stand directly beneath the highest part
- describe in broad terms how300
- Label right angled triangles with respect to a given angle.
- Use trigonometric ratios to calculate the length of opposite and adjacent sides, and the hypotenuse in right angled triangles.
- Use trigonometric ratios to calculate the size of angles in right angled triangles.
Maths skills required300