The value of sec 60° is 2. This is derived from the fact that cos 60° = 1/2 and sec is the reciprocal of cosine: sec 60° = 1/cos 60° = 1/(1/2) = 2. The complete set of trigonometric values at 60° are: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3, cot 60° = 1/√3, sec 60° = 2, cosec 60° = 2/√3 = 2√3/3.
sec 60° = 2 (exact value, no radicals).
Derived from cos 60° = 1/2 and sec = 1/cos: sec 60° = 1/(1/2) = 2.
cosec 60° = 2/√3 = 2√3/3 ≈ 1.1547.
cot 60° = 1/√3 = √3/3 ≈ 0.5774.
The 30-60-90 triangle has sides in ratio 1 : √3 : 2, from which all 60° values are derived.
sec²60° = 1 + tan²60° → 4 = 1 + 3 ✓ (Pythagorean identity verified).
In radians: 60° = π/3, so sec(π/3) = 2.
sec 60° = 2
Step-by-step derivation:
Verification using the 30-60-90 triangle: In a 30-60-90 right triangle with hypotenuse = 2:
For the 60° angle:
Summary: sec 60° = 2 (exact integer value)
Complete trig table for 60°:
| Function | Value | Simplified | Decimal |
|---|---|---|---|
| sin 60° | √3/2 | √3/2 | 0.8660 |
| cos 60° | 1/2 | 0.5 | 0.5 |
| tan 60° | √3 | √3 | 1.7321 |
| cot 60° | 1/√3 | √3/3 | 0.5774 |
| sec 60° | 2 | 2 | 2 |
| cosec 60° | 2/√3 | 2√3/3 | 1.1547 |
Notes:
Verification: sin²60° + cos²60° = (√3/2)² + (1/2)² = 3/4 + 1/4 = 1 ✓ Verification: tan 60° = sin 60°/cos 60° = (√3/2)/(1/2) = √3 ✓
The standard 30-60-90 triangle is the basis for all trig values at 60° (and 30°).
A 30-60-90 triangle has:
If the shortest side (opposite 30°) = 1:
For the 60° angle:
Note: The 30-60-90 triangle is half of an equilateral triangle with side 2. Cutting it in half gives a right triangle with hypotenuse 2, shorter leg 1, and longer leg √3.
| Angle | sin | cos | tan | sec | cosec | cot |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 1 | Undefined | Undefined |
| 30° | 1/2 | √3/2 | 1/√3 | 2/√3 | 2 | √3 |
| 45° | 1/√2 | 1/√2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2 | 2/√3 | 1/√3 |
| 90° | 1 | 0 | Undefined | Undefined | 1 | 0 |
Pattern for sec values: sec 0°=1, sec 30°=2/√3, sec 45°=√2, sec 60°=2, sec 90°=undefined
Notice that sec 60° = 2 is a clean integer, making it one of the most commonly tested values in exams.
On the unit circle at 60°:
Since sec θ = 1/cos θ: sec 60° = 1 / (1/2) = 2
Related identities involving sec 60°:
sec²60° = 1 + tan²60° (Pythagorean identity) → 2² = 1 + (√3)² → 4 = 1 + 3 = 4 ✓
sec 60° = cosec 30° → 2 = 2 ✓ (because 60° + 30° = 90°, complementary angles)
sec 60° × cos 60° = 1 → 2 × (1/2) = 1 ✓
In radians: 60° = π/3, so sec(π/3) = 2.
sec 60° = 2. This is because cos 60° = 1/2 and sec = 1/cos. So sec 60° = 1/(1/2) = 2.
sec 60° = 1/cos 60° = 1/(1/2) = 2. Alternatively, in a 30-60-90 triangle with hypotenuse 2 and adjacent side 1, sec 60° = hypotenuse/adjacent = 2/1 = 2.
cosec 60° = 1/sin 60° = 1/(√3/2) = 2/√3 = 2√3/3 ≈ 1.1547. To rationalise: multiply by √3/√3 to get 2√3/3.
cot 60° = 1/tan 60° = 1/√3 = √3/3 ≈ 0.5774. Rationalised form: multiply by √3/√3 to get √3/3.
sec 60° = 2 and sec 30° = 2/√3. They are not equal. However, sec 60° = cosec 30° = 2 (because 60° and 30° are complementary angles, and sec(90°−θ) = cosec θ).
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