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B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Exact Values of All Six Ratios, Including Coterminal and Quadrantal Angles

B1.4 B1.2About 65 minutes
Learning goals
  • I can find exact values of csc, sec and cot for special angles from 0 to 2π.
  • I can find exact values at the quadrantal angles and say when a ratio is undefined.
  • I can find exact values for angles greater than 2π and for negative angles, using coterminal angles.
  • I can evaluate expressions that mix all six ratios exactly.

Words to know

reciprocal ratios
csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ = cos θ/sin θ.
undefined
A ratio is undefined at an angle where it would require dividing by zero, such as tan(π/2) = 1/0.
coterminal angles
Angles that share a terminal arm, such as π/6 and 13π/6. Their ratios are equal.
principal angle
The coterminal angle between 0 and 2π. It is the easiest one to work with.

Flip the fraction

Once you know sin, cos and tan exactly, the reciprocal ratios take one step: flip the fraction and rationalize the denominator if needed.

  • sin(π/6) = 1/2, so csc(π/6) = 2.
  • cos(π/6) = √3/2, so sec(π/6) = 2/√3 = 2√3/3.
  • tan(π/6) = 1/√3, so cot(π/6) = √3.

The sign never changes when you flip: a reciprocal has the same sign as the ratio it comes from.

θcsc θsec θcot θ
π/622√3/3√3
π/4√2√21
π/32√3/32√3/3
Reciprocal ratios of the special angles in quadrant I. Use the related acute angle and CAST for the other quadrants, exactly as before.

Quadrantal angles: where ratios can be undefined

At the quadrantal angles the terminal arm lies on an axis, and one coordinate of the unit-circle point is 0.

  • At π/2 and 3π/2, x = cos θ = 0, so tan θ = y/x and sec θ = 1/x are undefined, while cot θ = x/y = 0.
  • At 0, π and 2π, y = sin θ = 0, so cot θ and csc θ are undefined, while tan θ = 0.
θ = 0 or 2π: (1, 0)θ = π/2: (0, 1)θ = π: (−1, 0)θ = 3π/2: (0, −1)x = 0 here, sotan and sec areundefined at π/2, 3π/2y = 0 here, socot and csc areundefined at 0, π, 2π
The four quadrantal points on the unit circle. A ratio with x in the denominator is undefined where x = 0; a ratio with y in the denominator is undefined where y = 0.
θsincostancscseccot
0010undefined1undefined
π/210undefined1undefined0
π0−10undefined−1undefined
3π/2−10undefined−1undefined0
All six ratios at the quadrantal angles. The values at 2π are the same as at 0.

Angles beyond 2π, and negative angles

Coterminal angles share a terminal arm, so they share every ratio. To find an exact value for any angle:

  1. Add or subtract multiples of 2π until the angle is between 0 and 2π (the principal angle).
  2. Use the related acute angle and CAST as usual.

For example, 13π/6 − 2π = 13π/6 − 12π/6 = π/6, and −3π/4 + 2π = 5π/4.

terminal arm13π/6 = 2π + π/6One full turn, then π/6 more.Same terminal arm as π/6,so the same ratios:sin(13π/6) = sin(π/6) = 1/2
The angle 13π/6 makes one full turn and then π/6 more, so it ends on the same terminal arm as π/6.
Worked example

Example 1: A reciprocal ratio in quadrant II

Find the exact value of sec(3π/4).

  1. 3π/4 is in quadrant II with β = π/4.
  2. cos(3π/4) = −√2/2 (cosine is negative in quadrant II).
  3. sec(3π/4) = 1 ÷ (−√2/2) = −2/√2 = −√2.

sec(3π/4) = −√2

Worked example

Example 2: An angle greater than 2π

Find the exact value of csc(19π/6).

  1. 19π/6 − 2π = 19π/6 − 12π/6 = 7π/6.
  2. 7π/6 is in quadrant III with β = π/6, so sin(7π/6) = −1/2.
  3. csc(19π/6) = csc(7π/6) = 1 ÷ (−1/2) = −2.

csc(19π/6) = −2

Worked example

Example 3: A negative angle

Find the exact value of cot(−2π/3).

  1. −2π/3 + 2π = 4π/3 (quadrant III, β = π/3).
  2. tan(4π/3) = √3 (tangent positive in quadrant III).
  3. cot(−2π/3) = 1/√3 = √3/3.

cot(−2π/3) = √3/3

Worked example

Example 4: Mixing all six

Find the exact value of sec²(π/6) − cot(5π/4) + csc(3π/2).

  1. sec(π/6) = 2/√3, so sec²(π/6) = 4/3.
  2. 5π/4: quadrant III, β = π/4, tan = 1, so cot(5π/4) = 1.
  3. csc(3π/2) = 1 ÷ sin(3π/2) = 1 ÷ (−1) = −1.
  4. 4/3 − 1 + (−1) = 4/3 − 2 = −2/3.

−2/3

Just for fun

Why did sec(π/2) never get invited to parties? Because every time someone asked for its value, it said "undefined."

Hands-on activity

Flash-card speed round

You need: 16 index cards (or pieces of paper cut to card size, with scissors used carefully), a pen, a calculator and a family member or friend.

  1. On the front of each card write one angle: 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7π/6, 5π/4, 4π/3, 3π/2, 5π/3, 7π/4, 11π/6.
  2. On the back, write all six exact ratios (write "undefined" where needed). Check each one with the calculator in radian mode, using 1 ÷ for the reciprocals. (For an undefined ratio, the calculator will give an error, or a huge number because π is rounded.)
  3. Your helper shuffles the cards, holds one up and names a ratio ("sec!"). Answer exactly. They flip the card to check.
  4. Make it harder: your helper adds 2π to the angle out loud ("π/3 plus 2π: cot?"). The answer must stay the same.

What to look for: answers that come from reasoning (quadrant, β, sign, flip) rather than guessing. Aim for 16 correct in a row.

Your turn

1

Find the exact value of csc(π/6).

2

Find the exact value of sec(5π/4).

3

Find the exact value of cot(5π/3).

4

Find the exact value of csc(3π/2).

5

Find sec(π/2), or explain why it does not exist.

6

Find the exact value of sin(13π/6).

7

Find the exact value of cos(−2π/3).

More practice: warm-up

1

sec 0 = ?

2

cot(π/4) = ?

3

csc(π/2) = ?

4

sec(π/3) = ?

5

cot(π/2) = ?

More practice: on your own

1

Find the exact value of tan(−π/4).

2

Find the exact value of csc(17π/6).

3

Find the exact value of sec²(π/6) − tan²(π/6).

4

Which of the six ratios are undefined at θ = π? Explain.

5

Find the exact value of cot(11π/4).

Challenge

1

Find the exact value of csc²(2π/3) + sec²(2π/3).

2

List every angle θ with 0 ≤ θ ≤ 2π where sec θ is undefined, and every one where cot θ is undefined.

3

Show that cot(π/3) = tan(π/6). Use the 30-60-90 triangle to explain why cot θ = tan(π/2 − θ) for acute θ.

Check your understanding

Quiz

1.

What is the exact value of csc(π/4)?

2.

What is the exact value of sec(2π/3)?

3.

Which of these is undefined?

4.

What is the exact value of cos(25π/6)?

5.

What is the exact value of cot(π/6)?

6.

Which angle between 0 and 2π is coterminal with −5π/6?

For parents and teachers

Big idea (B1.4): reciprocal ratios are found by flipping the exact value of sin, cos or tan. At the quadrantal angles one coordinate is 0, so some ratios are undefined. Angles outside 0 to 2π are first moved to a coterminal angle.

How to help: Use the flash-card game in the activity. When your teen says a ratio is undefined, ask "which coordinate is zero, and is it in the denominator?"

Common difficulties:

  • Writing 0 instead of undefined (or the other way round): cot(π/2) = 0 but tan(π/2) is undefined.
  • Forgetting to rationalize: 2/√3 should become 2√3/3.
  • Subtracting π instead of 2π to find a coterminal angle.