Pure mathematics
Trigonometry: Radians, Identities and Advanced Formulae
Year 12 · Year 13
- ✓By the end of this lesson students will be able to convert between radians and degrees, and use radians to calculate arc length and sector area.
- ✓By the end of this lesson students will be able to use and prove a range of trigonometric identities, including reciprocal, Pythagorean, compound angle, and double angle formulae.
- ✓By the end of this lesson students will be able to solve trigonometric equations using identities and advanced formulae, giving solutions in specified ranges.
- ✓By the end of this lesson students will be able to express expressions of the form a sinx + b cosx in the form Rsin(x±α) or Rcos(x±α), and use this form to solve equations and find maximum/minimum values.
Key concepts
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. It is a unit of angular measure, particularly useful in calculus and advanced trigonometry. The conversion between radians and degrees is based on the fact that a full circle (360°) is equal to 2π radians.
When the angle θ is measured in radians, simple formulae can be used to calculate the length of an arc and the area of a sector of a circle with radius r.
Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables for which both sides of the equation are defined. They are fundamental for simplifying expressions and solving equations.
These formulae allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. They are crucial for deriving other identities and solving more complex equations.
Derived directly from the compound angle formulae by setting A=B, these identities express trigonometric functions of 2A in terms of functions of A. They are particularly useful for simplifying expressions and solving equations involving 2x.
Any expression of the form a sinx + b cosx can be rewritten as Rsin(x+α) or Rcos(x±α), where R > 0 and α is an acute angle. This form is useful for finding maximum/minimum values, solving equations, and sketching graphs.
Key facts to remember
- 1π radians = 180°. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
- 2For angle θ in radians: Arc Length L = rθ, Area of Sector A = (1/2)r²θ.
- 3Fundamental identities: tanθ ≡ sinθ/cosθ and sin²θ + cos²θ ≡ 1.
- 4Reciprocal identities: secθ ≡ 1/cosθ, cosecθ ≡ 1/sinθ, cotθ ≡ 1/tanθ. Pythagorean identities: 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ.
- 5Compound angle formulae: sin(A±B) ≡ sinAcosB ± cosAsinB, cos(A±B) ≡ cosAcosB ∓ sinAsinB, tan(A±B) ≡ (tanA ± tanB) / (1 ∓ tanAtanB).
- 6Double angle formulae: sin(2A) ≡ 2sinAcosA, cos(2A) ≡ cos²A - sin²A ≡ 2cos²A - 1 ≡ 1 - 2sin²A, tan(2A) ≡ 2tanA / (1 - tan²A).
- 7The expression a sinx + b cosx can be written as Rsin(x+α) where R = √(a² + b²) and tanα = b/a (with careful consideration of α's quadrant).
- 8The maximum value of Rsin(x+α) is R, and the minimum value is -R.
Worked examples
Example 1
A sector of a circle has radius 10 cm and an angle of 1.8 radians. Calculate the arc length and the area of the sector. Give your answers to 3 significant figures.
Answer
Arc length = 18.0 cm (3 s.f.) Area of sector = 90.0 cm² (3 s.f.)
Ensure your calculator is in radian mode if you were to evaluate trigonometric functions of 1.8 radians, though not needed for these specific formulae.
Example 2
Solve the equation 2cos²x + sinx = 1 for 0 ≤ x < 2π, giving your answers in radians to 3 significant figures.
Answer
x = 1.57, 3.67, 5.76 radians (3 s.f.)
Always check the specified range for x and ensure your calculator is in radian mode when finding principal values.
Example 3
a) Express 4sinx + 3cosx in the form Rsin(x+α), where R > 0 and 0 < α < 90°. Give α to one decimal place. b) Hence, find the maximum value of 4sinx + 3cosx and the smallest positive value of x for which it occurs.
Answer
a) 4sinx + 3cosx ≡ 5sin(x + 36.9°) b) Maximum value = 5, occurs when x = 53.1°.
Ensure α is in the correct quadrant based on the signs of 'a' and 'b'. In this case, both 4 and 3 are positive, so α is acute (0 < α < 90°).
Common mistakes
- ✗Using degree mode on a calculator when angles are in radians, or vice-versa, especially when solving equations or using arc/sector formulae.
- ✗Algebraic errors when manipulating trigonometric identities, such as incorrectly squaring terms or failing to factorise.
- ✗Forgetting to consider all possible solutions within the given range for trigonometric equations, often by not using the CAST diagram or sketching the graph.
- ✗Incorrectly determining the angle α in the Rsin(x+α) form, particularly when 'a' or 'b' are negative, leading to α being in the wrong quadrant.
- ✗Confusing the compound angle formulae, especially the signs in cos(A±B) and tan(A±B).
Exam tips
- ★Always check your calculator's angle mode (radians or degrees) before starting any trigonometry question.
- ★When proving identities, work on one side at a time, transforming it into the other side. Do not treat it as an equation by moving terms across the equals sign.
- ★For trigonometric equations, find the principal value first, then use the CAST diagram or sketch the graph of the function to find all solutions within the specified range.
- ★When using the Rsin(x+α) form, clearly show the steps for finding R and α, and remember that R must be positive.
- ★Practise converting between the different forms of cos(2A) as they are frequently used to simplify expressions or solve equations.
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.