Mathematics • Year 9 • Unit 4 • Lesson 2
Identities in the Real World
Use the Pythagorean identity, the ratio identity tan = sin/cos, and complementary angles in real contexts — solar panels, GPS triangles, navigation bearings, kite strings, and a screen tilt. Then explain your method in your own words.
1. Word problems
Each problem uses one of the identities from Lesson 2: sin²θ + cos²θ = 1, tan θ = sin θ / cos θ, or complementary angles sin θ = cos(90° − θ). Show your working. Calculator in degree mode where needed.
1.1 — Solar panel angle. A solar panel installer in Wollongong tells a customer that the panel's cosine of the tilt angle from vertical equals 0.8. The customer wants to know the sine of that same angle (used in a different efficiency formula).
(a) Use the Pythagorean identity to find sin θ (assume θ is acute).
(b) Find the tilt angle θ to the nearest degree. 3 marks
1.2 — GPS triangle. A phone's GPS uses three satellites. The angle from the phone to one satellite has sin = 12/13. The phone's app instead needs cos and tan of the same angle.
(a) Find cos θ and tan θ without finding θ itself. 3 marks
1.3 — Navigation bearing. A ship's bearing is given as "S 35° E". A navigator notices that the cosine of 35° is the same as the sine of another angle.
(a) Use complementary angles to state which angle has the same sine as cos 35°.
(b) Verify with a calculator (4 dp): cos 35° = sin ____° = ____. 3 marks
1.4 — Kite string angles. A kite is up in the air. The string makes an angle θ with the ground where tan θ = 7/4 (the kite is 7 m up for every 4 m across).
(a) Find sin θ and cos θ as fractions in lowest terms (use Pythagoras to find the hypotenuse first).
(b) Verify sin²θ + cos²θ = 1 from your fractions. 3 marks
1.5 — Screen tilt. A laptop screen is tilted so that sin θ = 0.5, where θ is the angle the screen makes with the keyboard (so the screen is vertical when θ = 90°, and flat when θ = 0°).
(a) Find θ.
(b) Without using sin or cos, find tan θ exactly (use exact values).
(c) State the COMPLEMENTARY angle (between the screen and the vertical). 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes: "sin² 30° + cos² 30° = sin 30° × 2 + cos 30° × 2 = 1 + √3 = 2.73." They got the right answer was supposed to be 1, but they're confused. In your own words, explain (i) what mistake they made when they wrote sin² 30° as sin 30° × 2, (ii) what sin² 30° actually means, and (iii) work the calculation correctly to show the answer really is 1. Refer to "the square of sin θ" somewhere.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Solar panel angle
(a) sin²θ = 1 − cos²θ = 1 − 0.8² = 1 − 0.64 = 0.36 → sin θ = √0.36 = 0.6.
(b) θ = cos⁻¹(0.8) ≈ 37° (or sin⁻¹(0.6) ≈ 36.87°).
This is the 3-4-5 right-angled triangle in disguise: cos = 4/5 = 0.8, sin = 3/5 = 0.6.
1.2 — GPS triangle
cos²θ = 1 − (12/13)² = 1 − 144/169 = 25/169 → cos θ = 5/13.
tan θ = sin θ / cos θ = (12/13) / (5/13) = 12/5.
This is the 5-12-13 Pythagorean triple.
1.3 — Navigation bearing
(a) cos 35° = sin(90° − 35°) = sin 55°.
(b) Calculator check: cos 35° ≈ 0.8192, sin 55° ≈ 0.8192. ✓
1.4 — Kite string
Build the triangle: opposite = 7, adjacent = 4. By Pythagoras hypotenuse = √(49 + 16) = √65.
sin θ = opp/hyp = 7/√65 (or 7√65/65 rationalised).
cos θ = adj/hyp = 4/√65 (or 4√65/65 rationalised).
Check: sin²θ + cos²θ = 49/65 + 16/65 = 65/65 = 1. ✓
1.5 — Screen tilt
(a) sin θ = 0.5 → θ = sin⁻¹(0.5) = 30°.
(b) tan 30° = sin 30° / cos 30° = (1/2) / (√3/2) = 1/√3 = √3/3.
(c) Complementary angle = 90° − 30° = 60°.
So the angle between the screen and the vertical is 60°.
2.1 — Explain your thinking (sample response)
My classmate has confused sin² 30° with 2 × sin 30°. The square of sin θ (written sin²θ) means (sin θ)², which is sin θ MULTIPLIED BY ITSELF — not multiplied by 2. The correct working uses the exact values from Lesson 1: sin 30° = 1/2 and cos 30° = √3/2. So sin² 30° = (1/2)² = 1/4 and cos² 30° = (√3/2)² = 3/4. Adding: 1/4 + 3/4 = 4/4 = 1. ✓ The identity sin²θ + cos²θ = 1 only works when "²" really means "squared" (multiplied by itself).
Marking: 1 mark for identifying the mistake (× 2 vs squared); 1 for defining sin² θ as (sin θ)²; 1 for the correct numerical working showing 1/4 + 3/4 = 1; 1 for a clear, full-sentence explanation.