KnowledgeMadeFun, home

Lesson · Grade 10 · Foundations of Mathematics and Pre Calculus

Angles of Elevation and Depression: Measuring What You Cannot Reach

  • Grade 10
  • Foundations of Mathematics and Pre Calculus
  • 2 min 12 s
  • British Columbia
  • BC10-MA-38

Standing 20 m from a tree with eyes 1.5 m up, a 40° angle of elevation gives h = 20 × tan 40° ≈ 16.8 m above the eyes. Adding the eye height makes the tree 18.3 m, which is checked with 16.8 ÷ 20 = 0.84. Looking down from a cliff with eyes 30 m above the sea at a boat 50 m out, tan θ = 30 ÷ 50 = 0.6 gives an angle of depression of about 31.0°, and the video ends on a one-line recap: measure from the horizontal and use tan = opposite over adjacent. It points out the two common mistakes: forgetting the eye height, and measuring from the vertical, which gives 59.0°.

Anyone can watch two lessons on the website without an account. Signing in is free, no payment and no card, and opens three a day; Plus has no limit.

Loading

Share to Google Classroom

In one sentence

The question this lesson answers

How tall is a tree you cannot climb? One angle can tell you.

You stand 20 metres from the tree, with your eyes 1.5 metres above the ground.

Transcript

What the lesson says

How tall is a tree you cannot climb? One angle can tell you.

You stand 20 metres from the tree, with your eyes 1.5 metres above the ground.

Your eye level is horizontal. The angle up from it to your line of sight to the treetop is the angle of elevation, 40 degrees.

The height above your eyes, h, is across from the 40 degree angle: the opposite side. The 20 metre side beside it is adjacent.

For an angle theta in a right triangle, the tangent, tan theta, is opposite over adjacent.

So tan 40 degrees is h over 20. Multiply both sides by 20: h is 20 times tan 40 degrees.

With a calculator in degree mode, tan 40 degrees is about 0.8391, so h is about 16.8 metres.

Forgetting your eye height is a common mistake. Add it: 16.8 plus 1.5 is 18.3 metres.

Check: 16.8 divided by 20 is 0.84, which matches tan 40 degrees.

Now look down from a cliff. Your eyes are 30 metres above the sea, and a boat is 50 metres out.

The angle down from the horizontal to your line of sight is the angle of depression, theta.

The 30 metre height is opposite theta and the 50 metres is adjacent, so tan theta is 30 over 50, or 0.6.

Inverse tan turns 0.6 back into an angle: about 31.0 degrees. Check: tan 31.0 degrees is about 0.601.

Measuring from the vertical is also a common mistake: it gives 59.0 degrees.

Up or down, measure from the horizontal, then use tan: opposite over adjacent.

Curriculum

Where this fits

British Columbia, Grade 10

Foundations of Mathematics and Pre Calculus 10, Trigonometry: primary trigonometric ratiosBC10-MA-38

Previous: Finding a Missing Angle with Inverse TanNext: Solving a Right Triangle: Pythagoras and Trig

British Columbia, Grade 11

Workplace Mathematics 11, Rates and slope: rate of changeBC11-MA-91

This lesson is also listed in Grade 11.

The outcomes and strands follow the British Columbia curriculum.

Keep going

More on this topic