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AI HL · 12.10

Geometry & Trigonometry

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12.10.1

Right-angled trigonometry and angles between lines

12.10.1

Right-angled trigonometry and angles between lines

Target: I can find an angle or a side of a right-angled triangle with sine, cosine or tangent, choosing the ratio from the sides I have. (+3 more)

By the end of this section

Key points

Right-angled triangles

  • Label the sides first: the hypotenuse is opposite the right angle; the opposite and adjacent sides are named from the angle you are using.
  • \(\sin\theta=\frac{\text{opp}}{\text{hyp}},\quad \cos\theta=\frac{\text{adj}}{\text{hyp}},\quad \tan\theta=\frac{\text{opp}}{\text{adj}}\). Use the ratio that has the two sides you know or want.
  • To find an angle, use the inverse function: sin⁻¹, cos⁻¹ or tan⁻¹. Check the calculator is in degrees.
  • When the unknown side is the hypotenuse, it ends up underneath: \(x=\frac{9.5}{\sin 37°}\). The hypotenuse is always the longest side.

The angle of a line

  • A line with gradient m has a triangle with run 1 and rise m. So \(\tan\theta=m\) and θ = tan⁻¹ m is the angle with the positive x-axis.
  • The x-axis and the y-axis are 90° apart, so the angle with the y-axis is 90° − θ.
  • For a negative gradient, ignore the minus sign when you find the angle: lengths are positive. Sketch the line to see which angle you have.
  • "Acute" means less than 90°. If you get an obtuse angle, subtract it from 180°.

The angle between two lines

  • Sketch both lines. Find the angle each makes with the horizontal, using tan⁻¹ of the gradient.
  • Both gradients positive (or both negative): subtract the two angles.
  • One positive and one negative: add the two angles. If the sum is more than 90°, the acute angle is 180° minus the sum.
  • Only the gradients matter. The y-intercepts move the lines but do not change the angle between them.

On the calculator

Degree mode and inverse functions

  • Set the calculator to degrees before any triangle question.
  • Use sin⁻¹, cos⁻¹ or tan⁻¹ to get an angle from a ratio.
  • Type the ratio as a fraction, or keep the full calculator value, so you do not round too early.

Screenshot and video to come

Angle between two lines

  • Find each tan⁻¹ of a gradient and keep both full values.
  • Subtract or add them on the calculator, then round once at the end.

Screenshot and video to come

Explore: the angle between two lines

Each line goes through two points you can drag. Find the angle each line makes with the x-axis, then the acute angle between the lines.

Acute angle between the lines

The dashed steps show each gradient: run across, then rise up. Click a point, then use the arrow keys to move it by half a unit.

Drag a point to change a line.

  1. Make both lines rise with different gradients. Do the two angles add or subtract? Why?
  2. Make one line rise and the other fall. Now what happens? When is the sum of the two angles more than 90°, and what do you do then?
  3. Move P and Q together so the first line keeps its gradient. What happens to the angle between the lines? What does that tell you about the y-intercept?
  4. Make the lines parallel. What is the angle between them? What is true about their gradients?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
An angle from two sides

Triangle PQR has a right angle at Q. PQ = 8.4 cm and PR = 13.0 cm. Find angle QPR.

13.0 cm8.4 cmθQPR
Working
  1. Label the sides

    PQ is adjacent to P. PR is the hypotenuse.

    The hypotenuse is opposite the right angle.

  2. Pick the ratio

    \(\cos P=\frac{8.4}{13.0}\)

    Adjacent and hypotenuse: cosine.

  3. Use the inverse

    From the calculator: P = cos⁻¹(8.4 ÷ 13.0) = 49.7°

    Use the inverse function to get an angle. The calculator must be in degrees.

Video walkthrough coming
A hypotenuse

Triangle XYZ has a right angle at Y. Angle X = 37° and YZ = 9.5 cm. Find the length of XZ.

x9.5 cm37°YXZ
Working
  1. Label the sides

    YZ is opposite X. XZ is the hypotenuse.

    Opposite and hypotenuse: sine.

  2. Write the equation

    \(\sin 37°=\frac{9.5}{x}\)

    The unknown is underneath.

  3. Rearrange

    \(x=\frac{9.5}{\sin 37°}\)

    Multiply both sides by x, then divide by sin 37°.

  4. Calculate

    From the calculator: x = 15.8 cm

    The hypotenuse is the longest side: it is longer than 9.5 cm.

Video walkthrough coming
A line and the x-axis

Find the angle between the line y = 1.5x and the positive x-axis.

−11234−1123456xy
Working
  1. Gradient triangle

    Run 1, rise 1.5

    A gradient of 1.5 means up 1.5 for every 1 across.

  2. Tangent

    \(\tan\theta=\frac{1.5}{1}\)

    Opposite over adjacent.

  3. Inverse

    From the calculator: θ = tan⁻¹(1.5) = 56.3°

    In general the angle with the positive x-axis is tan⁻¹ of the gradient.

Video walkthrough coming
A line and the y-axis

Find the angle between the line y = 4x and the positive y-axis.

−1−0.50.511.522.532468xy
Working
  1. Angle with the x-axis

    From the calculator: tan⁻¹(4) = 76.0°

    Use a gradient triangle with run 1 and rise 4.

  2. Take it from 90°

    90° − 76.0° = 14.0°

    The axes meet at 90°. The line sits between them.

Video walkthrough coming
A downhill line and the y-axis

Find the acute angle between the line y = −1.6x and the y-axis.

−4−3−2−1123−3−2−112345xy
Working
  1. Ignore the minus

    Use rise 1.6 and run 1

    The minus only tells you the line slopes down to the right. Lengths are positive.

  2. Angle with the x-axis

    From the calculator: tan⁻¹(1.6) = 58.0°

    This is the acute angle with the x-axis.

  3. Angle with the y-axis

    90° − 58.0° = 32.0°

    Sketch the line: the acute angle with the y-axis is on the same side.

Video walkthrough coming
Two lines through the origin

Find the acute angle between the lines y = 4x and y = 1.5x.

−0.50.511.522.532468xy
Working
  1. Sketch

    Both lines slope up. Each makes an angle with the x-axis.

    The angle between them is the difference of these two angles.

  2. The steeper line

    α = tan⁻¹(4) = 76.0°

    From the calculator.

  3. The other line

    β = tan⁻¹(1.5) = 56.3°

    From the calculator.

  4. Subtract

    θ = 76.0° − 56.3° = 19.7°

    Use the unrounded values on the calculator to keep 3 significant figures.

Video walkthrough coming
Lines with intercepts

Find the acute angle between the lines y = 3x − 2 and y = 5 − 0.5x.

−112345−4−22468xy
Working
  1. Only the gradients matter

    Gradients 3 and −0.5

    The intercepts move the lines but do not change the angle between them.

  2. Angle of each line

    α = tan⁻¹(3) = 71.6°, β = tan⁻¹(0.5) = 26.6°

    Ignore the minus sign. β is the angle the falling line makes with the horizontal.

  3. Add them

    α + β = 98.1°

    One line rises and one falls, so the two angles add up to the angle across the gap.

  4. Acute angle

    θ = 180° − 98.1° = 81.9°

    Angles on a straight line add to 180°. Give the acute one.

Video walkthrough coming