Trigonometry Cheat Sheet 2026

The 30 highest-yield Trigonometry facts, distilled from real exam questions. Print it, save it as a PDF, or study it here — free, no sign-up.

1. The word 'trigonometry' derives from Greek roots meaning 'triangle' and what other concept? → Measure
2. What is the relationship between theory and practice in Inverse Functions? → Theory provides the foundation; practice applies it to real situations
3. In right triangle PQR with the right angle at R, if tan P = 3/4, what is sin P? → 3/5
4. Who introduced the modern notation for trigonometric functions, using abbreviations like 'sin' and 'cos'? → Leonhard Euler
5. Evaluate: Find θ in [0, 2π) where the polar curve r = sin(2θ) passes through the pole. → θ = 0, π/2, π, 3π/2
6. Which identity represents the product-to-sum formula for sin A · cos B? → (1/2)[sin(A+B) + sin(A−B)]
7. A marine biologist models the depth of a diving submarine as d(t) = 200 − 150 cos(π t / 10) feet, where t is minutes. What is the maximum depth reached? → 350 feet
8. Solve sin(x) + cos(x) = 1 for x in [0, 2π). → x = 0, π/2
9. Which 15th-century Islamic astronomer compiled the most accurate trigonometric tables of the medieval period at the Samarkand Observatory? → Ulugh Beg
10. Which identity shows that sin(−θ) = −sin θ? → Sine is an odd function
11. The inverse function arctan(x) has what range? → (-π/2, π/2)
12. What is the polar equation of the vertical line x = 3? → r cos θ = 3
13. A surveyor measures a plot of triangular land with two sides of 200 m and 150 m enclosing an angle of 72°. What is the area of the plot? → 14,266 m²
14. Which polar equation represents a circle of radius 5 centered at the origin? → r = 5
15. What is correlation? → A statistical relationship between two variables
16. Which of the following equals sin(A + B)? → sin A cos B + cos A sin B
17. What is the range of y = 4sin(x) − 1? → [−5, 3]
18. Which study approach is most effective for Polar Coordinates material? → Active recall with practice questions
19. Which 9th-century Islamic mathematician's work 'Kitab al-fusul fi al-hisab al-Hindi' helped transmit Indian trigonometric methods to the Islamic world? → Al-Khawarizmi
20. What was the primary purpose of trigonometry in ancient India? → Astronomy
21. In a right triangle with a 45° angle, if the hypotenuse is 10, what are the lengths of the two legs? → 5√2 each
22. In triangle RST, R = 65°, r = 18, s = 15. Applying the Law of Sines, what is sin S? → sin S = 15·sin 65° / 18
23. What does a pie chart represent? → Parts of a whole as percentages
24. Nicolaus Copernicus included trigonometric methods in his heliocentric model primarily because he needed them to: → Calculate angular positions of planets from observational data
25. At the pole, the curve r = sin θ cos θ has tangent lines. How many distinct tangent lines exist there? → 2
26. Convert r = 2/(3 − sin θ) to rectangular form and identify the conic. → Ellipse with e = 1/3
27. What is the range of the data set: 3, 7, 1, 9, 5? → 8
28. What is the reciprocal of the sine function? → Cosecant
29. A triangular garden has sides of 40 ft, 55 ft, and 70 ft. Which expression gives the cosine of the largest angle? → (40² + 55² − 70²) / (2·40·55)
30. What is the midline of y = 2cos(x) − 5? → y = −5