Trigonometric Identities
This topic covers two fundamental trigonometric identities that link sine, cosine, and tangent. They are essential tools for simplifying trigonometric expressions and solving equations without a calculator.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The identity sin2 θ + cos2 θ = 1 is a direct consequence of Pythagoras's theorem applied to a unit circle, and it holds true for any angle θ.
- The identity tan θ = sin θ / cos θ is derived from the basic definitions (SOH CAH TOA) of the trigonometric ratios.
- Use sin2 θ + cos2 θ = 1 to find the value of sin θ if you know cos θ, or vice versa. Remember to consider the correct sign based on the angle's quadrant.
- Use tan θ = sin θ / cos θ to find the tangent of an angle when sine and cosine are known, or to rewrite an expression in terms of sine and cosine only.
- These identities are often used together in multi-step problems to find one trigonometric ratio from another.
Formulae
sin2 θ + cos2 θ = 1 When you know either sin θ or cos θ and need to find the other. Also used to simplify expressions containing squared trigonometric functions.
tan θ = sin θ / cos θ When you need to find tan θ from sin θ and cos θ, or when you need to express tan θ in terms of sin θ and cos θ to simplify an equation.
Definitions
- Pythagorean Identity
- The relationship sin2 θ + cos2 θ = 1. It relates the square of the sine of an angle to the square of its cosine.
- Quotient Identity
- The relationship tan θ = sin θ / cos θ. It defines the tangent of an angle as the ratio of its sine to its cosine.
Worked example
Given that sin x = -12/13 and that 270° < x < 360°, find the exact value of tan x.
- 1
Start with the Pythagorean identity to find cos x:
sin2 x + cos2 x = 1.
- 2
Substitute the given value of sin x:
(-12/13)2 + cos2 x = 1.
- 3
Calculate the square:
144/169 + cos2 x = 1.
- 4
Rearrange to solve for cos2 x:
cos2 x = 1 - 144/169 = 25/169 - 5
Take the square root:
cos x = ±√(25/169) = ±5/13 - 6
Determine the sign of cos x.
The range 270° < x < 360° is the fourth quadrant, where cosine is positive.
Therefore, cos x = +5/13.
- 7
Now use the quotient identity:
tan x = sin x / cos x - 8
Substitute the known values:
tan x = (-12/13) / (5/13) - 9
Simplify the fraction:
tan x = -12/5
Answer: -12/5
Common mistakes
- ×Forgetting the ± when taking the square root to find sin θ or cos θ. You must use the angle's quadrant (CAST diagram) to determine the correct sign.
- ×Making an algebraic slip when rearranging the Pythagorean identity, for example incorrectly writing sin θ = 1 - cos θ.
- ×Ignoring the condition that tan θ is undefined when cos θ = 0 (at 90°, 270°, etc.).
No-calculator tips
- ✓Look for Pythagorean triples (3-4-5, 5-12-13, etc.) when using sin2 θ + cos2 θ = 1. If you see sin θ = 12/13, you can immediately deduce the magnitude of cos θ must be 5/13, saving calculation time.
- ✓To divide fractions like (a/c) / (b/c), you can cancel the common denominator 'c' directly to get a/b. This is quicker than multiplying by the reciprocal.
- ✓Always draw a quick mental or physical sketch of the CAST diagram to double-check the signs of sin, cos, and tan in the relevant quadrant. This prevents easy-to-make sign errors.