Grade 10 — Mathematics 10-3 › T3U07 Primary Trigonometric Ratios

Grade 10 › T3U07 Primary Trigonometric Ratios › L3
GRADE 10 · ALBERTAOUTCOME AB-MAT10-3-G4

L3 · Finding an angle; solving job-site problems

Learning goal

“I can work the ratios backwards - compute a ratio from two known sides and read off the angle it belongs to - and I can solve complete job-site problems (angles of elevation and depression, heights by indirect measurement), judging whether the answer is reasonable.”

Angle of elevation — the angle between the horizontal and the line of sight when looking UP at an object.
Angle of depression — the angle between the horizontal and the line of sight when looking DOWN at an object; it equals the angle of elevation from the object looking up (alternate interior angles).
Line of sight — the straight line from the observer's eye to the object; it is the hypotenuse of the working triangle.
Inverse operation (for angles) — undoing a ratio to recover the angle: given sin = 0.8, the inverse step (the calculator's inverse sine key) returns the angle, about 53°.
Worked example

In a right triangle, the side opposite angle θ is 8 m and the hypotenuse is 10 m. Find θ. (Given: the angle whose sine is 0.8 is about 53°.) Step 1 (choose): opposite + hypotenuse = SINE. Step 2 (compute the ratio from the sides): sin θ = 8/10 = 0.8. Step 3 (invert): θ is the angle whose sine is 0.8, so θ is about 53°. Say it plainly: finding an angle is two moves - divide the sides to get the ratio, then run the ratio backwards (the calculator's inverse key) to get the angle. The ratio you compute must be < 1 for sine and cosine; if it isn't, the fraction is upside down. Landmark values worth knowing cold: sin 30° = 0.5; cos 60° = 0.5; tan 45° = 1 (opposite = adjacent).

Check yourself — explain it

You know the opposite and adjacent sides and want the angle. Your first move is...

Paired problem — example still visible

Adjacent 6 m, hypotenuse 12 m. Find the angle. (The angle whose cosine is 0.5 is 60°.) -> cos = 6/12 = 0.5, so the angle is 60°

Faded completion — last step is yours

"Opposite 3 m, hypotenuse 6 m: sin = 3/6 = ___, so the angle = ___°." -> 0.5; 30°

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