Grade by Grade
B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Trigonometric Ratios of Angles in Radians, with Technology

B1.3About 65 minutes
Learning goals
  • I can set my calculator to radian mode and check that it is in the right mode.
  • I can find sin θ, cos θ and tan θ for any angle given in radians, with a calculator.
  • I can find csc θ, sec θ and cot θ in radians using reciprocals.
  • I can predict the sign of each ratio from the quadrant, using the CAST rule with radian boundaries.

Words to know

radian mode
The calculator setting (often shown as RAD or R) in which angles typed into sin, cos and tan are read as radians.
unit circle
The circle with centre (0, 0) and radius 1. For an angle θ in standard position, its terminal arm meets the circle at (cos θ, sin θ).
cosecant (csc)
The reciprocal of sine: csc θ = 1/sin θ.
secant (sec)
The reciprocal of cosine: sec θ = 1/cos θ.
cotangent (cot)
The reciprocal of tangent: cot θ = 1/tan θ = cos θ/sin θ.
CAST rule
A memory aid for which ratios are positive in each quadrant: All (I), Sine (II), Tangent (III), Cosine (IV).

Same ratios, new unit

The trigonometric ratios do not care what unit you use for the angle. An angle of π/3 and an angle of 60° are the same angle, so sin(π/3) = sin 60°. What changes is how you type the angle into a calculator.

Every scientific calculator has a mode. In degree mode, sin 2 means the sine of 2°. In radian mode, it means the sine of 2 radians, which is a much bigger angle (about 115°).

  • In degree mode, sin 2 ≈ 0.0349.
  • In radian mode, sin 2 ≈ 0.9093.

Mode check: type cos(π). Radian mode gives −1. If you get 0.9985, you are in degree mode. (On most phones, turn the calculator sideways and look for a "Rad/Deg" key.)

Remember from Grade 11: ratios from a point

If the terminal arm of θ passes through the point (x, y), and r = √(x² + y²) is its distance from the origin, then

sin θ = y/r, cos θ = x/r, tan θ = y/x

and the reciprocal ratios are

csc θ = r/y, sec θ = r/x, cot θ = x/y.

On the unit circle (r = 1), the point where the terminal arm meets the circle is simply (cos θ, sin θ).

xyP(cos 2, sin 2)2 rad−0.420.9111On the unit circle (r = 1):x-coordinate = cos θy-coordinate = sin θtan θ = y ÷ xθ = 2 is in quadrant II,so cos 2 < 0 and sin 2 > 0.
The angle 2 rad ends in quadrant II. Its terminal arm meets the unit circle at about (−0.42, 0.91), so cos 2 ≈ −0.42 and sin 2 ≈ 0.91.

CAST in radians

The signs of the ratios depend only on the quadrant, so the CAST rule still works. Only the boundary labels change: 0, π/2 ≈ 1.57, π ≈ 3.14, 3π/2 ≈ 4.71, 2π ≈ 6.28. Learn these decimals so you can place an angle like θ = 4 (quadrant III) instantly.

Quadrant I: 0 < θ < π/2All ratios positiveQuadrant II: π/2 < θ < πSine (and csc) positiveQuadrant III: π < θ < 3π/2Tangent (and cot) positiveQuadrant IV: 3π/2 < θ < 2πCosine (and sec) positiveπ/2π3π/20, 2π
The CAST rule with radian boundaries. A reciprocal ratio always has the same sign as the ratio it comes from.

Reciprocal is not inverse

Calculators have no csc, sec or cot keys. Use 1 ÷ sin, 1 ÷ cos and 1 ÷ tan. The key marked sin⁻¹ is not cosecant: it is the inverse sine, which answers "what angle has this sine?"

Worked example

Example 1: Primary ratios of a radian angle

Find sin 2, cos 2 and tan 2 to four decimal places. Check the signs with CAST.

  1. Set the calculator to radian mode.
  2. sin 2 ≈ 0.9093, cos 2 ≈ −0.4161, tan 2 ≈ −2.1850.
  3. Check: π/2 ≈ 1.57 < 2 < π ≈ 3.14, so 2 is in quadrant II, where only sine is positive ✓.

sin 2 ≈ 0.9093, cos 2 ≈ −0.4161, tan 2 ≈ −2.1850

Worked example

Example 2: Reciprocal ratios

Find csc 4, sec 4 and cot 4 to four decimal places.

  1. In radian mode: sin 4 ≈ −0.7568, cos 4 ≈ −0.6536, tan 4 ≈ 1.1578.
  2. csc 4 = 1 ÷ sin 4 ≈ 1 ÷ (−0.7568) ≈ −1.3213.
  3. sec 4 = 1 ÷ cos 4 ≈ −1.5299.
  4. cot 4 = 1 ÷ tan 4 ≈ 0.8637.
  5. Check: 4 is between π and 3π/2, quadrant III, where only tangent and cotangent are positive ✓. (Use the unrounded values on the calculator; dividing by a rounded value can change the last digit.)

csc 4 ≈ −1.3213, sec 4 ≈ −1.5299, cot 4 ≈ 0.8637

Worked example

Example 3: An angle written with π

Find sin(5π/7), cos(5π/7) and tan(5π/7) to four decimal places.

  1. Type the angle with brackets: sin(5π/7). Without brackets, some calculators compute (sin 5) × π ÷ 7.
  2. sin(5π/7) ≈ 0.7818, cos(5π/7) ≈ −0.6235, tan(5π/7) ≈ −1.2540.
  3. Check: 5π/7 is between π/2 = 3.5π/7 and π = 7π/7, so it is in quadrant II ✓.

0.7818, −0.6235, −1.2540

Worked example

Example 4: From a point to all six ratios and the angle

The point (−3, −4) is on the terminal arm of θ, where 0 ≤ θ < 2π. Find all six ratios and find θ to two decimal places.

  1. r = √((−3)² + (−4)²) = √25 = 5.
  2. sin θ = −4/5, cos θ = −3/5, tan θ = −4/−3 = 4/3.
  3. csc θ = −5/4, sec θ = −5/3, cot θ = 3/4.
  4. The point is in quadrant III. The related acute angle is β = tan⁻¹(4/3) ≈ 0.9273 (radian mode).
  5. In quadrant III, θ = π + β ≈ 3.1416 + 0.9273 ≈ 4.0689.

sin θ = −4/5, cos θ = −3/5, tan θ = 4/3, csc θ = −5/4, sec θ = −5/3, cot θ = 3/4; θ ≈ 4.07.

QuadrantAngle from related acute angle β
Iθ = β
IIθ = π − β
IIIθ = π + β
IVθ = 2π − β
The Grade 11 rules for related acute angles, with 180° replaced by π and 360° by 2π.
Just for fun

A calculator in degree mode says sin 2 = 0.0349. It is not wrong. It just answered a question you did not ask: the sine of 2 degrees.

Hands-on activity

Find the angle of your stairs in radians

You need: a tape measure or ruler, a calculator in radian mode, and a staircase at home (or the steps to your door). Have a grown-up nearby, and stay off busy stairs.

  1. Measure the rise of one step (its height) and the run (the depth of the tread you step on), in centimetres.
  2. The stairs slope at an angle θ with tan θ = rise ÷ run. Find θ = tan⁻¹(rise ÷ run) in radian mode.
  3. Switch to degree mode and find θ again. Check that (radian answer) × 180/π matches the degree answer.
  4. Use the angle to find sin θ, cos θ and csc θ in radian mode.
  5. Measure along the slope from the front edge of one step to the front edge of the next (the hypotenuse h). Check that sin θ ≈ rise ÷ h.

What to look for: many home stairs rise about 18 to 20 cm per step with a tread of about 23 to 28 cm, so θ is often about 0.6 to 0.7 rad (roughly 35° to 40°). Your exact values depend on your stairs; what matters is that the radian and degree answers agree.

Your turn

1

Find sin 1.3 to four decimal places.

2

Find cos 5 to four decimal places.

3

Find tan(3π/8) to four decimal places.

4

Find csc 0.5 to four decimal places.

5

Find sec 2.8 to four decimal places.

6

Find cot(4π/5) to four decimal places.

7

Without a calculator, predict the signs of sin 3, cos 3 and tan 3.

More practice: warm-up

1

Which calculator mode do you need for sin 1.2?

2

In which quadrant does θ = 4 lie?

3

Complete: csc θ = 1 ÷ ___.

4

What is cos 0?

5

Is tan θ positive or negative in quadrant III?

More practice: on your own

1

The point (5, −12) is on the terminal arm of θ, 0 ≤ θ < 2π. Find all six ratios and θ to two decimal places.

2

Evaluate 3 sin 1 + 2 cos 1 to two decimal places.

3

Find csc 0.4. A friend pressed sin⁻¹ 0.4 and got 0.4115. What went wrong?

4

Without a calculator, decide which is larger: sin 1 or sin 2. Explain, then check.

5

cos θ = −0.3 and θ is in quadrant III. Find θ to two decimal places.

Challenge

1

Use a calculator to check that sin²(2.7) + cos²(2.7) = 1 and that 1 + tan²(2.7) = sec²(2.7). Why should these be true for every angle?

2

The terminal arm of θ passes through (−2, 7), 0 ≤ θ < 2π. Find r, sin θ, sec θ and θ to two decimal places.

3

For which angles θ with 0 < θ < 2π are sin θ and cos θ both negative and |sin θ| > |cos θ|? Explain.

Check your understanding

Quiz

1.

Your calculator says sin(π/2) = 0.0274. What is most likely wrong?

2.

csc 1 is closest to...

3.

An angle of 5 radians in standard position ends in...

4.

For π/2 < θ < π, which ratio is positive?

5.

sec θ is equal to...

6.

cot 0.8, to three decimal places, is...

For parents and teachers

Big idea (B1.3): the ratios are the same as in Grade 11; only the angle unit changes. The calculator must be in radian mode, and csc, sec and cot are found with 1 ÷ sin, 1 ÷ cos and 1 ÷ tan.

How to help: Before homework, ask "What is cos π on your calculator?" If it is not −1, the mode is wrong. Ask your teen to predict the sign of an answer from the quadrant before pressing any keys.

Common difficulties:

  • Degree mode by accident (the most common error on tests).
  • Pressing sin⁻¹ for cosecant.
  • Typing sin 5π/7 without brackets.
  • Forgetting the quadrant boundaries in decimals (1.57, 3.14, 4.71, 6.28).