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Sine, cosine and tangent

In every right triangle with a 35° angle, the side opposite that angle is about 0.574 of the longest side, whether the triangle is drawn on a napkin or across a valley.

Naming the sides from an angle

Trigonometry starts with a right triangle and one of its two acute angles, usually called θ (theta). The longest side, across from the right angle, is the hypotenuse. The other two sides get their names from θ. The side touching θ that is not the hypotenuse is the adjacent side, from the Latin for "lying next to". The side across the triangle from θ is the opposite side. Pick the other acute angle instead and the names of these two sides swap, which OpenStax points out explicitly: the side adjacent to one acute angle is opposite the other.

Seen from θ: a is the opposite side, b the adjacent side, c the hypotenuse. The names always depend on the chosen angle.

Three ratios

Divide one side by another and you get a ratio. Three of these ratios have names. The sine of θ is opposite over hypotenuse. The cosine of θ is adjacent over hypotenuse. The tangent of θ is opposite over adjacent. OpenStax gives the memory aid SohCahToa: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. In a triangle with sides 3, 4 and 5, the angle opposite the side 4 has sin = 4/5 = 0.8, cos = 3/5 = 0.6 and tan = 4/3, about 1.333.

sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.

Why the angle alone decides the ratio

Draw two right triangles with the same 35° angle, one small and one large. Their three angles match (35°, 90° and 55°), so they are similar: the large one is a scaled copy of the small one. Scaling multiplies every side by the same number, and that number cancels in any ratio of two sides. So sin 35° is a single number, about 0.5736, no matter how big the triangle is. That is what lets a calculator store it. In the lab, keep θ fixed and drag the hypotenuse slider: every length changes, the ratios do not.

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A right triangle on a grid of unit squares. Drag the top corner or use the sliders; choose sin, cos or tan to light up the two sides of that ratio.

A calculator gives these values directly, but check that it is set to degrees (often shown as DEG). In radian mode, sin 30 returns about −0.988 instead of 0.5, because it reads the 30 as 30 radians. OpenStax also defines three reciprocal ratios: secant = hypotenuse / adjacent, cosecant = hypotenuse / opposite and cotangent = adjacent / opposite. They are the first three turned upside down, so this course works with sine, cosine and tangent.

BackgroundMacTutor traces trigonometry to astronomy. The first known table for it was a table of chords, the straight line across a circle cut off by a given angle, produced by the Greek astronomer Hipparchus in about 140 BC. His tables have not survived, but he is said to have written twelve books of them, which is why MacTutor calls him the founder of trigonometry. The word tangent came much later: the Danish mathematician Thomas Fincke first used it in 1583.
  1. Which side is the hypotenuse?
    • the side opposite the right angle
    • the side next to θ
    • the side opposite θ
    • the shortest side
  2. Sides 5, 12, 13. Angle θ is opposite the side 5. sin θ to 4 decimal places?
  3. Same triangle. cos θ to 4 decimal places?
  4. Same triangle. tan θ as a fraction?
  5. OpenStax example: adjacent side 15, hypotenuse 17. cos α?
  6. A triangle with sides 6, 8, 10 compared with one with sides 3, 4, 5. For matching angles, the sine is:
    • the same
    • twice as big
    • half as big
    • four times as big
  7. Hypotenuse 6 and θ = 35°. Length of the opposite side, to 2 decimal places?
  8. tan 35° to 4 decimal places?
  9. sin 30 on a calculator gives −0.988. What is wrong?
    • it is in radian mode
    • the battery is low
    • sine can be negative
    • nothing
  10. In a right triangle, sin θ is always:
    • between 0 and 1
    • more than 1
    • equal to tan θ
    • negative
  11. Sides 3, 4, 5 with θ opposite the 4. sec θ = hypotenuse / adjacent?
Key points
  • Hypotenuse: across from the right angle. Opposite and adjacent depend on θ.
  • sin = opp/hyp, cos = adj/hyp, tan = opp/adj (SohCahToa).
  • Similar triangles give the same ratios, so each angle has one sine, cosine and tangent.
  • Set the calculator to degrees.
  • sec, csc, cot are the reciprocals of cos, sin, tan.