Triangle Basics — “Trig or Pythagorean"
A right triangle has legs of 3 cm and 4 cm. You need to find the hypotenuse.
→ Do you use Trig or the Pythagorean Theorem?
Pythagorean Theorem — both legs are known, and you’re finding the hypotenuse.
Define sine, cosine, and tangent for an acute angle in a right triangle.
sin=opp/hyp
cos = adj/hyp
tan = opp/adj
Find the hypotenuse of a triangle with legs 9 cm and 12 cm.
15 cm
A 10 ft ladder leans against a wall at a 60° angle. How high up the wall does it reach?
10 sin 60° ≈ 8.7 ft
What makes two triangles similar?
Same shape — corresponding angles equal, sides proportional.
A right triangle has a leg of 8 cm and a hypotenuse of 10 cm. You need to find one of the acute angles.
Trig — use sine, cosine, or tangent to find the angle.
If sin A = 4/5, what sides of the triangle does this represent?
Opposite = 4, Hypotenuse = 5
Find the missing leg if hypotenuse = 13 cm and one leg = 5 cm.
12 cm
A kite string makes a 40° angle with the ground and is 50 m long. Find the height of the kite.
50 sin 40° ≈ 32.1 m
How can you tell right triangles with the same acute angle are similar?
They share the same angle measures.
You know one leg and the hypotenuse, and you need to find the missing leg.
→ Trig or Pythagorean Theorem?
Either could work, but Pythagorean is more direct if finding the side.
Write a trig equation for finding the angle in a right triangle where the opposite side is 6 and adjacent is 10.
tan θ = 6/10
Find the angle when opposite = 7 cm and adjacent = 24 cm.
θ = tan⁻¹(7/24)
θ≈ 16°
A ramp rises 3 ft over a horizontal distance of 12 ft. Find the angle of elevation.
θ = tan⁻¹(3/12)
θ ≈ 14°
Why are sin 30° and cos 60° equal?
Because 30° and 60° are complementary angles.
≈ Because 30° and 60° add up to 90°
You know one side length and one acute angle (not 90°). You need to find another side.
→ Trig or Pythagorean Theorem?
Trig — you need sine, cosine, or tangent because not all sides are known.
In a triangle, cos x = 0.6. What ratio of sides does this describe?
Adjacent / Hypotenuse = 0.6
Find the height of a tree if its shadow is 20 m long and the angle of elevation to the top is 35°.
h = 20 tan 35°
h≈ 14.0 m
A plane descends at an angle of 5° from 3000 ft. How far (horizontally) does it travel before landing?
3000 / tan 5° ≈ 34,300 ft
Explain why trigonometric ratios don’t depend on the triangle’s size.
All similar right triangles have proportional sides.
A ladder leans against a wall forming a right triangle with the ground. You know the height the ladder reaches and the angle it makes with the ground. Find the ladder’s length.
→ Trig or Pythagorean Theorem?
Trig — use cosine or sine depending on which side is known.
A triangle has sin A = 0.6428. Estimate the measure of angle A.
A ≈ 40°
Find all side lengths of a triangle if one angle = 25°, hypotenuse = 18.
opp = 18 sin 25° opp≈ 7.6,
adj = 18 cos 25° ≈ 16.3
Astronomers observe a parallax angle of 0.00002°. Estimate the distance (AU) to the star using d = 1 / tan p.
d ≈ 1 / tan (0.00002°)
d ≈ 2.9 × 10⁶ AU
Prove that tan θ = sin θ / cos θ using right triangle definitions.
tan = (opp/adj); sin / cos = (opp/hyp)/(adj/hyp) = opp/adj.
You’re designing a wheelchair ramp that rises 2.5 feet over a distance of 10 feet. The ramp will also have a handrail that extends diagonally from the top of the ramp to the ground.
Which method should you use to find the length of the handrail — Trig or the Pythagorean Theorem — and why?
Then, calculate the handrail’s length.
Pythagorean Theorem — two sides of the right triangle are known (2.5 ft rise, 10 ft run).
Handrail = √(2.5² + 10²) ≈ 10.3 ft.
In a right triangle, angle A = 32°, and the adjacent side is 9 m.
a) Write a trig equation to find the hypotenuse.
b) Solve for the hypotenuse and round to the nearest tenth.
a) cos(32°) = 9 / h
b) h = 9 / cos(32°) ≈ 10.6 m
A surveyor measures the angle of elevation to the top of a building as 28° from a point 50 m away. Later, they move 30 m closer and find the new angle to be 41°.
Find the height of the building.
Set up two equations using tan(28°) = h / 50 and tan(41°) = h / 20, then solve for h.
h ≈ 26.5 m.
A rescue helicopter hovers directly above a mountain slope. The pilot looks down at a 35° angle of depression and sees a hiker 800 m away (measured along the ground).
How high is the helicopter above the hiker?
h = 800 × tan(35°) ≈ 560 m.
Two right triangles are similar. In the smaller triangle, sin A = 0.6. The larger triangle’s adjacent side to angle A is 15 cm.
Find the larger triangle’s hypotenuse and explain how similarity guarantees the result is valid.
sin A = opp/hyp → 0.6 = opp/hyp → hyp = opp/0.6.
By similarity, side ratios are constant, so the relationship holds regardless of triangle size.
If adjacent = 15, then opp = 15 × tan(sin⁻¹(0.6)) ≈ 12, hyp ≈ 20 cm.