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🇨🇩 Trigonometry

Unit circle and ratios

⏱ 30 minOfficial programme: CD-H1-TRI-01

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🎯 In this lesson, you will learn to

  • Use degrees, grades, radians on the unit circle.
  • Define and relate the six trigonometric ratios.

The lesson

Introduction

A surveyor in Kolwezi measures angles with an instrument set to grads, a student in Matadi uses degrees, and her calculator also offers radians. To switch between units and define the trigonometric ratios of any angle, we use the unit circle. In this lesson you learn to convert angles and to define and relate sine, cosine, tangent, cotangent, secant and cosecant.

Explanation

Angle units

A full turn measures 360∘360^\circ, or 400400 grads (400400 gr), or 2π2\pi radians. To convert, use
D180=G200=Rπ\frac{D}{180}=\frac{G}{200}=\frac{R}{\pi}
where DD, GG and RR are the measures of the same angle in degrees, grads and radians.

Degrees30∘30^\circ45∘45^\circ60∘60^\circ90∘90^\circ180∘180^\circ
Grads33.3‾33.\overline{3}505066.6‾66.\overline{6}100100200200
Radiansπ6\dfrac\pi6π4\dfrac\pi4π3\dfrac\pi3π2\dfrac\pi2π\pi

One radian is the central angle that cuts off an arc whose length equals the radius; 11 rad ≈57.3∘\approx57.3^\circ.

Explanation

Unit circle

It is the circle with centre OO and radius 11 in an orthonormal coordinate system, oriented in the positive direction (anticlockwise). To an angle α\alpha we associate the point MM of the circle such that the angle from [Ox)[Ox) to [OM)[OM) measures α\alpha. By definition:
cos⁡α=x-coordinate of M,sin⁡α=y-coordinate of M.\cos\alpha=\text{x-coordinate of }M,\qquad\sin\alpha=\text{y-coordinate of }M.

So −1≤cos⁡α≤1-1\le\cos\alpha\le1 and −1≤sin⁡α≤1-1\le\sin\alpha\le1. The axes split the circle into four quadrants:

Quadrant1st2nd3rd4th
cos⁡α\cos\alpha++−-−-++
sin⁡α\sin\alpha++++−-−-

By Pythagoras in the triangle formed by OO, MM and its projection on (Ox)(Ox): cos⁡2α+sin⁡2α=1\cos^2\alpha+\sin^2\alpha=1.

Explanation

The six trigonometric ratios

  • tan⁡α=sin⁡αcos⁡α\tan\alpha=\dfrac{\sin\alpha}{\cos\alpha} (if cos⁡α≠0\cos\alpha\ne0); cot⁡α=cos⁡αsin⁡α\cot\alpha=\dfrac{\cos\alpha}{\sin\alpha} (if sin⁡α≠0\sin\alpha\ne0).
  • sec⁡α=1cos⁡α\sec\alpha=\dfrac1{\cos\alpha} (if cos⁡α≠0\cos\alpha\ne0); csc⁡α=1sin⁡α\csc\alpha=\dfrac1{\sin\alpha} (if sin⁡α≠0\sin\alpha\ne0).
  • Relations: tan⁡α⋅cot⁡α=1\tan\alpha\cdot\cot\alpha=1; 1+tan⁡2α=sec⁡2α=1cos⁡2α1+\tan^2\alpha=\sec^2\alpha=\dfrac1{\cos^2\alpha}; 1+cot⁡2α=csc⁡2α1+\cot^2\alpha=\csc^2\alpha.

Special values

α\alpha0030∘30^\circ45∘45^\circ60∘60^\circ90∘90^\circ
sin⁡α\sin\alpha0012\dfrac1222\dfrac{\sqrt2}232\dfrac{\sqrt3}211
cos⁡α\cos\alpha1132\dfrac{\sqrt3}222\dfrac{\sqrt2}212\dfrac1200
tan⁡α\tan\alpha0033\dfrac{\sqrt3}3113\sqrt3undefined

Worked example

Problem: Convert 135∘135^\circ to grads and radians, then 7070 gr to degrees.

Step 1: G=200×135180=150G=\dfrac{200\times135}{180}=150 gr.

Step 2: R=π×135180=3π4R=\dfrac{\pi\times135}{180}=\dfrac{3\pi}4 rad.

Step 3: D=180×70200=63∘D=\dfrac{180\times70}{200}=63^\circ.

Answer: 135∘=150135^\circ=150 gr =3π4=\dfrac{3\pi}4 rad and 7070 gr =63∘=63^\circ.

Worked example

Problem: The angle α\alpha is in the 2nd quadrant and sin⁡α=35\sin\alpha=\dfrac35. Compute the other five ratios.

Step 1: cos⁡2α=1−925=1625\cos^2\alpha=1-\dfrac9{25}=\dfrac{16}{25}. In the 2nd quadrant, cos⁡α<0\cos\alpha<0, so cos⁡α=−45\cos\alpha=-\dfrac45.

Step 2: tan⁡α=3/5−4/5=−34\tan\alpha=\dfrac{3/5}{-4/5}=-\dfrac34 and cot⁡α=−43\cot\alpha=-\dfrac43.

Step 3: sec⁡α=1cos⁡α=−54\sec\alpha=\dfrac1{\cos\alpha}=-\dfrac54 and csc⁡α=1sin⁡α=53\csc\alpha=\dfrac1{\sin\alpha}=\dfrac53.

Step 4: Check: 1+tan⁡2α=1+916=2516=sec⁡2α1+\tan^2\alpha=1+\dfrac9{16}=\dfrac{25}{16}=\sec^2\alpha.

Answer: cos⁡α=−45\cos\alpha=-\dfrac45, tan⁡α=−34\tan\alpha=-\dfrac34, cot⁡α=−43\cot\alpha=-\dfrac43, sec⁡α=−54\sec\alpha=-\dfrac54, csc⁡α=53\csc\alpha=\dfrac53.

Worked example

Problem: Compute the exact value of A=sin⁡230∘+cos⁡60∘⋅tan⁡45∘+sec⁡60∘A=\sin^2 30^\circ+\cos60^\circ\cdot\tan45^\circ+\sec60^\circ.

Step 1: sin⁡30∘=12\sin30^\circ=\dfrac12, so sin⁡230∘=14\sin^2 30^\circ=\dfrac14.

Step 2: cos⁡60∘⋅tan⁡45∘=12×1=12\cos60^\circ\cdot\tan45^\circ=\dfrac12\times1=\dfrac12.

Step 3: sec⁡60∘=1cos⁡60∘=2\sec60^\circ=\dfrac1{\cos60^\circ}=2.

Answer: A=14+12+2=114A=\dfrac14+\dfrac12+2=\dfrac{11}4.

Watch out

Watch out: The secant is the reciprocal of the cosine and the cosecant the reciprocal of the sine: sec⁡α=1cos⁡α\sec\alpha=\dfrac1{\cos\alpha}, not 1sin⁡α\dfrac1{\sin\alpha}. After using cos⁡2α+sin⁡2α=1\cos^2\alpha+\sin^2\alpha=1, choose the sign according to the quadrant. On a calculator, check the mode (DEG, RAD or GRAD) before any computation.

Key points

  • D180=G200=Rπ\dfrac{D}{180}=\dfrac{G}{200}=\dfrac{R}{\pi}; one turn =360∘=400=360^\circ=400 gr =2π=2\pi rad.
  • On the unit circle, M(cos⁡α,sin⁡α)M(\cos\alpha, \sin\alpha) and cos⁡2α+sin⁡2α=1\cos^2\alpha+\sin^2\alpha=1.
  • tan⁡=sin⁡cos⁡\tan=\dfrac{\sin}{\cos}, cot⁡=cos⁡sin⁡\cot=\dfrac{\cos}{\sin}, sec⁡=1cos⁡\sec=\dfrac1{\cos}, csc⁡=1sin⁡\csc=\dfrac1{\sin}.
  • 1+tan⁡2α=sec⁡2α1+\tan^2\alpha=\sec^2\alpha; 1+cot⁡2α=csc⁡2α1+\cot^2\alpha=\csc^2\alpha.
  • The sign depends on the quadrant.

✍️ Exercises

Answer one exercise at a time. Your answer is checked right away, with an explanation.

Exercise 1 of 8

★ Easy

Convert 45∘45^\circ to grads.

gr

You can use a decimal comma or point (2,5 or 2.5).