SATHEE: Trigonometry Values

Trigonometry Values - Quick Revision

sin 0° = 0, sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = ½, cos 90° = 0
tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = ∞
cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
sin²θ + cos²θ = 1 → Pythagorean identity
1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
sin (90° – θ) = cos θ; cos (90° – θ) = sin θ
tan (90° – θ) = cot θ; cot (90° – θ) = tan θ
sin (–θ) = –sin θ; cos (–θ) = cos θ; tan (–θ) = –tan θ
0°, 30°, 45°, 60°, 90° → remember root-pattern: 0-1-2-3-4 divided by 2 for sin; reverse for cos
sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ – sin²θ
Maximum value of sin θ & cos θ is 1; minimum is –1
tan θ = sin θ / cos θ; cot θ = cos θ / sin θ
θ increases in 1st quadrant → sin, cos & tan all increase
Always rationalise denominators (e.g., 1/√2 → √2/2) in final answers

Formula/Rule	Application
sin (A ± B) = sin A cos B ± cos A sin B	Compound-angle quick expansion
cos (A ± B) = cos A cos B ∓ sin A sin B	Same as above for cosine
tan (A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)	Combine two angles into one tan
sin 2θ = 2 sin θ cos θ	Double-angle, height-distance problems
cos 2θ = 2 cos²θ – 1 = 1 – 2 sin²θ	Express cos² or sin² in terms of cos 2θ
sin θ = Opposite/Hypotenuse	Right-triangle definition
cos θ = Adjacent/Hypotenuse	Right-triangle definition
tan θ = Opposite/Adjacent	Right-triangle definition
sec θ = 1/cos θ; cosec θ = 1/sin θ; cot θ = 1/tan θ	Reciprocal identities
sin²θ + cos²θ = 1	Fundamental Pythagorean identity

“0-1-2-3-4” rule: write √0/2, √1/2, √2/2, √3/2, √4/2 → gives sin 0° to 90° directly.
“SIP” – Sin Increases, Cos Decreases in 0°-90°.
“Pandit Badri Prasad” – P-B-P / H-H-H → sin 30°=½, cos 30°=√3/2, tan 30°=1/√3.
Reciprocals: CO-SEC is SEC of CO-mplement; CO-TAN is TAN of CO-mplement.
ASTC – “All School Teachers Cry” (All, Sin, Tan, Cos positive in I, II, III, IV quadrants).

Mistake	Correct Approach
Writing tan 90° = 0	tan 90° is undefined (∞)
Forgetting to rationalise 1/√2	Always write √2/2 in final answer
Mixing sin (A + B) with cos (A + B) signs	Use “Sin keeps sign, Cos changes sign” rule
Using θ in degrees but calculator in radians	Check mode (Deg/Rad) every time
Taking sec θ = 1/tan θ	sec θ = 1/cos θ; cosec θ = 1/sin θ

Glance at the 0°-90° table right before entry—5 sec visual scan locks values.
Solve 2 height-distance problems to warm-up; they use tan θ most.
Write identities on rough sheet immediately after getting answer sheet—saves 5 mins later.
If option contains both √3/2 & 2/√3 → recall cos 30° vs sec 30° to pick correct one.
Any expression with 90°±θ → use “Complementary” rule first, reduces calculation 50%.

1. What is the value of sin 120°?

**Ans:** sin 120° = sin (180°–60°) = sin 60° = √3/2

2. If tan θ = 3/4, then sec θ is?

**Ans:** sec θ = √(1 + tan²θ) = √(1 + 9/16) = √(25/16) = 5/4

3. cos 15° equals

**Ans:** cos (45°–30°) = cos 45° cos 30° + sin 45° sin 30° = (√2/2)(√3/2) + (√2/2)(½) = (√6 + √2)/4

4. Maximum value of 5 sin θ + 12 cos θ is

**Ans:** √(5² + 12²) = 13

5. tan 225° = ?

**Ans:** 225° = 180° + 45° → tan positive in III quad; tan 225° = tan 45° = 1