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D. Trigonometric Functions · Trigonometric Ratios, Identities and Laws

Reciprocal Trigonometric Ratios: Cosecant, Secant and Cotangent

D1.4 D1.2About 65 minutes
Learning goals
  • I can define cosecant, secant and cotangent in a right triangle using its sides.
  • I can relate csc, sec and cot to sin, cos and tan as reciprocals.
  • I can find exact and approximate values of the reciprocal ratios for angles from 0° to 360°.
  • I can solve simple equations involving csc, sec and cot.

Words to know

reciprocal
The reciprocal of a non-zero number n is 1/n. A fraction's reciprocal is found by flipping it: the reciprocal of 3/5 is 5/3.
cosecant (csc)
csc θ = hypotenuse/opposite = r/y = 1/sin θ.
secant (sec)
sec θ = hypotenuse/adjacent = r/x = 1/cos θ.
cotangent (cot)
cot θ = adjacent/opposite = x/y = 1/tan θ = cos θ/sin θ.

Three more ratios

A right triangle has three sides, so you can make six different ratios out of them. You already know three: sine, cosine and tangent. The other three are their upside-down versions, called the reciprocal ratios. They show up in surveying, in physics, and especially in Grade 12 Calculus and Vectors (MCV4U), where the derivative of tan x is written with sec x.

Definitions in a right triangle

For an acute angle A in a right triangle:

  • cosecant: csc A = hypotenuse/opposite
  • secant: sec A = hypotenuse/adjacent
  • cotangent: cot A = adjacent/opposite

Compare each with its partner. The fraction is just flipped:

  • csc A = 1/sin A
  • sec A = 1/cos A
  • cot A = 1/tan A

Since tan A = sin A/cos A, flipping gives cot A = cos A/sin A as well.

AACBoppositehypotenuseadjacentsin A = opposite/hypotenusecsc A = hypotenuse/oppositecos A = adjacent/hypotenusesec A = hypotenuse/adjacenttan A = opposite/adjacentcot A = adjacent/oppositeTheir reciprocals:
The six ratios of angle A. Each one in the bottom group is the reciprocal (the flipped fraction) of the one directly above it in the top group.

Which goes with which?

Each pair has exactly one "co": sine pairs with cosecant, cosine pairs with secant, and tangent pairs with cotangent. So the "co" names do not go together: csc goes with sin, and sec goes with cos.

Worked example

Example 1: All six ratios

In right △ABC, ∠C = 90°, BC = 5, AC = 12 and AB = 13. Write all six trigonometric ratios of ∠A.

  1. For ∠A: opposite = BC = 5, adjacent = AC = 12, hypotenuse = AB = 13.
  2. sin A = 5/13, cos A = 12/13, tan A = 5/12.
  3. Flip each one: csc A = 13/5, sec A = 13/12, cot A = 12/5.

sin A = 5/13, cos A = 12/13, tan A = 5/12, csc A = 13/5, sec A = 13/12, cot A = 12/5

Angles from 0° to 360°

With a point P(x, y) on the terminal arm and r = √(x² + y²):

csc θ = r/y, sec θ = r/x, cot θ = x/y

A reciprocal ratio always has the same sign as its partner, so CAST still works: in quadrant II, sine and cosecant are positive and the other four are negative.

A reciprocal is undefined when its partner is 0. For example, sin 0° = 0, so csc 0° is undefined; cos 90° = 0, so sec 90° is undefined.

sin θ = y/rcsc θ = r/yflip the fractioncsc θ = 1/sin θcos θ = x/rsec θ = r/xflip the fractionsec θ = 1/cos θtan θ = y/xcot θ = x/yflip the fractioncot θ = 1/tan θ
Each reciprocal ratio is the flipped fraction of its partner, using a point (x, y) on the terminal arm at distance r from the origin.
Worked example

Example 2: Exact values

Find the exact values of csc 30°, sec 45°, cot 60° and sec 120°.

  1. csc 30° = 1/sin 30° = 1 ÷ (1/2) = 2.
  2. sec 45° = 1/cos 45° = 1 ÷ (1/√2) = √2.
  3. cot 60° = 1/tan 60° = 1/√3 = √3/3.
  4. sec 120° = 1/cos 120° = 1 ÷ (−1/2) = −2.

csc 30° = 2, sec 45° = √2, cot 60° = √3/3, sec 120° = −2

Worked example

Example 3: Using a calculator

Evaluate sec 125° to four decimal places.

  1. Most calculators have no sec key. Use sec 125° = 1/cos 125°.
  2. cos 125° ≈ −0.573 58.
  3. 1 ÷ (−0.573 58) ≈ −1.7434.

sec 125° ≈ −1.7434. Do not press cos⁻¹: that key finds an angle, not a reciprocal.

Worked example

Example 4: From a point

The point (−8, 15) is on the terminal arm of θ. Find csc θ, sec θ and cot θ.

  1. r = √(64 + 225) = √289 = 17.
  2. csc θ = r/y = 17/15, sec θ = r/x = 17/(−8) = −17/8, cot θ = x/y = −8/15.
  3. Check: θ is in quadrant II, so only sin and csc are positive. ✓

csc θ = 17/15, sec θ = −17/8, cot θ = −8/15

Worked example

Example 5: Solving an equation

Solve sec θ = −2.5 for 0° ≤ θ ≤ 360°, to the nearest tenth of a degree.

  1. Flip both sides: cos θ = 1/(−2.5) = −0.4.
  2. β = cos⁻¹(0.4) ≈ 66.4°.
  3. Cosine is negative in quadrants II and III: θ ≈ 180° − 66.4° = 113.6° or θ ≈ 180° + 66.4° = 246.4°.

θ ≈ 113.6° or 246.4°

Just for fun

Before electronic calculators, people looked up trigonometric values in printed books of tables, and many of those books listed secant and cosecant too. Having the reciprocals printed saved a lot of long division by hand. Today one press of the 1/x key does the same job.

Hands-on activity

Six ratios from a leaning board

You need: a long flat object such as a cutting board, a large hardcover book or a shelf board, a wall or a stack of books to lean it on, a tape measure and a calculator.

  1. Lean the board so it forms a right triangle with the floor (or table) and the wall.
  2. Measure the board's length (hypotenuse), the height where it touches the wall (opposite) and the distance from the wall to its foot (adjacent).
  3. Calculate all six ratios for the angle between the board and the floor.
  4. Multiply each pair: sin × csc, cos × sec, tan × cot.
  5. Use cos⁻¹(adjacent ÷ hypotenuse) to find the angle, then check sec of that angle with 1 ÷ cos.

What to look for: each product in step 4 should be very close to 1 (for example, 0.98 to 1.02), since reciprocals multiply to 1. Small differences come from measuring.

Your turn

1

A right triangle has sides 8, 15 and 17. The angle θ is opposite the side 8. Find csc θ, sec θ and cot θ.

2

Find the exact value of csc 45°.

3

Find the exact value of sec 60°.

4

Find the exact value of cot 30°.

5

Evaluate sec 200° to four decimal places.

6

Solve csc θ = 1.25 for 0° ≤ θ ≤ 360°, to the nearest tenth of a degree.

7

Find cot 90°, and explain why cot 0° is undefined.

More practice: warm-up

1

csc θ is the reciprocal of which ratio?

2

sec 0° = ?

3

csc 90° = ?

4

cot 45° = ?

5

sin θ = 0.25. What is csc θ?

More practice: on your own

1

The point (−5, −12) is on the terminal arm of θ. Find csc θ, sec θ and cot θ.

2

sec θ = 2 and θ is in quadrant IV. Find θ and the exact value of csc θ.

3

Sam pressed sin⁻¹ 0.5 to find csc 30° and got 30. Explain his mistake and give the correct value.

4

Find the exact value of csc²45° − cot²45°.

5

Solve cot θ = −2 for 0° ≤ θ ≤ 360°, to the nearest tenth of a degree.

Challenge

1

Explain why sec θ can never be between −1 and 1.

2

Solve sec²θ = 4 for 0° ≤ θ ≤ 360°.

3

A ramp rises 1 m for every 12 m of horizontal run. Let θ be its angle with the ground. Find cot θ and csc θ exactly, and find θ.

Check your understanding

Quiz

1.

csc A equals:

2.

In a right triangle, sec A equals:

3.

What is the exact value of sec 60°?

4.

What is cot 45°?

5.

The point (−3, 4) is on the terminal arm of θ. What is csc θ?

6.

Which of these is undefined?

For parents and teachers

Big idea (D1.4): cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent. In a right triangle they are the "flipped" side ratios (for example sec A = hypotenuse/adjacent).

How to help: Ask "What is csc paired with?" until the pairs are automatic (csc with sin, sec with cos, cot with tan). When your teen uses a calculator, watch which key they press: the reciprocal uses 1 ÷ (or x⁻¹ applied to the value), never the sin⁻¹ key.

Common difficulties:

  • Pairing csc with cos and sec with sin because of the "co" prefix.
  • Confusing the reciprocal 1/sin θ with the inverse sin⁻¹. On calculators the same symbol ⁻¹ is used for both ideas, which is confusing.
  • Forgetting that a reciprocal keeps the sign of its partner.