- Year 9
Tangent ratio
I can derive the tangent ratio from the sides of a right-angled triangle.
Lesson details
Key learning points
- When the adjacent has a length of one, the opposite side has a length of tan(θ).
- You can apply a scale factor to this triangle to find the scaled length of the opposite side.
- There is a multiplicative link between two similar right-angled triangles.
- There is a multiplicative link within each right-angled triangle.
Keywords
Adjacent - The adjacent side of a right-angled triangle is the side which is next to both the right angle and the marked angle.
Opposite - The opposite side of a right-angled triangle is the side which is opposite the marked angle.
Trigonometric ratios - The trigonometric ratios are ratios between each pair of lengths in a right-angled triangle.
Tangent function - The tangent of an angle (tan(θ°)) is the y-coordinate of point Q on the triangle which extends from the unit circle.
Common misconception
tan(60°) is double tan(30°).
The values of tan of an angle do not scale linearly. From the unit circle, we see that an angle of 30° meets a tangent to the circle at a height of approx. 0.58 units, whilst an angle of 60° meets the same tangent at a height of approx. 1.73 units.
Licence
Lesson video
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Prior knowledge starter quiz
6 Questions
Q1.In a right triangle, if the hypotenuse is 32.5 cm and a second side is 12.5 cm, what is the perimeter? (Use a calculator to help you.)
Q2.What is the value of θ for sin(θ) = tan(θ) ?
Q3.What is the value of θ for tan(θ) = 1
Q4.In a right triangle, if the hypotenuse is 32.5 cm and a second side is 12.5 cm, what is the area? (Use a calculator to help you.)
Q5.What is the value of θ for sin(θ) = 1
Q6.What is the value of θ for sin(θ) = cos(θ) ?
Assessment exit quiz
6 Questions
Q1.Match the estimated values with the trigonometric functions below.
sin(23°) - 0.39
cos(39°) - 0.777
tan(22°) - 0.404
tan(45°) - 1
Q2.For which value of θ is tan (θ) undefined?
Q3.In a right triangle, if you are provided with θ and the opposite side, which function would you use to find the adjacent side?
Q4.Which of these is a rearrangement of $$tan(θ)=\frac{opp}{adj}$$
Q5.Which of these is not an rearrangement of $$tan(θ)=\frac{opp}{adj}$$
Q6.For a right triangle ABC, if angle CBA = 33ᵒ, and the adjacent side to that angle is 1 cm, what is the length of the side opposite angle CBA?
