Trigonometry Beyond Right Triangles · Lesson 2

Trig values in every quadrant

Every angle in standard position ends in one of four quadrants. Its reference angle is the acute angle between the terminal side and the x-axis (never the y-axis). For θ\theta between 0∘0^\circ and 360∘360^\circ: in quadrant II the reference angle is 180∘−θ180^\circ - \theta, in quadrant III it is θ−180∘\theta - 180^\circ, and in quadrant IV it is 360∘−θ360^\circ - \theta.

A trig function at θ\theta has the same size as at the reference angle; only the sign can change. The sign comes from the quadrant, because cos⁡θ\cos\theta is the x-coordinate and sin⁡θ\sin\theta is the y-coordinate of the point on the unit circle.

Signs by quadrant
QuadrantAnglesPositiveNegative
I0∘0^\circ to 90∘90^\circsine, cosine, tangentnone
II90∘90^\circ to 180∘180^\circsinecosine, tangent
III180∘180^\circ to 270∘270^\circtangentsine, cosine
IV270∘270^\circ to 360∘360^\circcosinesine, tangent

When a question gives one ratio and the quadrant, draw a right triangle for the reference angle, find the third side with the Pythagorean theorem, and then attach the signs. For example, if cos⁡θ=−45\cos\theta = -\frac{4}{5} and θ\theta is in quadrant III, the triangle is 3-4-5, so sin⁡θ=−35\sin\theta = -\frac{3}{5} and tan⁡θ=34\tan\theta = \frac{3}{4}.

Coterminal angles share a terminal side. Add or subtract 360∘360^\circ (or 2π2\pi) as many times as you need: 75∘75^\circ, 435∘435^\circ, and −285∘-285^\circ are coterminal, so they have the same sine, cosine, and tangent.

Worked example Easy

What is the exact value of tan⁡240∘\tan 240^\circ?

  1. −3-\sqrt{3}
  2. −33-\frac{\sqrt{3}}{3}
  3. 33\frac{\sqrt{3}}{3}
  4. 3\sqrt{3}Answer

How to solve it

  1. 240∘240^\circ is between 180∘180^\circ and 270∘270^\circ, so it is in quadrant III.
  2. Reference angle: 240∘−180∘=60∘240^\circ - 180^\circ = 60^\circ, and tan⁡60∘=3\tan 60^\circ = \sqrt{3}.
  3. In quadrant III sine and cosine are both negative, so their quotient, the tangent, is positive.
  4. tan⁡240∘=3\tan 240^\circ = \sqrt{3}.

Why each choice is right or wrong

  • A. Incorrect. −3-\sqrt{3} has the right size but the wrong sign. In quadrant III sine and cosine are both negative, so tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta} is positive.
  • B. Incorrect. −33-\frac{\sqrt{3}}{3} measures the reference angle to the y-axis, 270∘−240∘=30∘270^\circ - 240^\circ = 30^\circ, and also gives the tangent the wrong sign. Measure to the x-axis: 60∘60^\circ.
  • C. Incorrect. 33\frac{\sqrt{3}}{3} is tan⁡30∘\tan 30^\circ, from measuring the reference angle to the y-axis. The reference angle is 240∘−180∘=60∘240^\circ - 180^\circ = 60^\circ, and tan⁡60∘=3\tan 60^\circ = \sqrt{3}.
  • D. Correct. The reference angle is 60∘60^\circ, so the size is tan⁡60∘=3\tan 60^\circ = \sqrt{3}, and the tangent is positive in quadrant III.

Worked example Medium

If sin⁡θ=513\sin\theta = \frac{5}{13} and 90∘<θ<180∘90^\circ < \theta < 180^\circ, what is the value of tan⁡θ\tan\theta?

  1. −125-\frac{12}{5}
  2. −1213-\frac{12}{13}
  3. −512-\frac{5}{12}Answer
  4. 512\frac{5}{12}

How to solve it

  1. Draw the reference triangle: opposite side 5, hypotenuse 13.
  2. The third side is 132−52=144=12\sqrt{13^2 - 5^2} = \sqrt{144} = 12.
  3. θ\theta is in quadrant II, where x is negative, so cos⁡θ=−1213\cos\theta = -\frac{12}{13}.
  4. tan⁡θ=sin⁡θcos⁡θ=513÷(−1213)=−512\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{5}{13} \div (-\frac{12}{13}) = -\frac{5}{12}.

Why each choice is right or wrong

  • A. Incorrect. −125-\frac{12}{5} is adjacent over opposite, the cotangent. The tangent is opposite over adjacent: −512-\frac{5}{12}.
  • B. Incorrect. −1213-\frac{12}{13} is cos⁡θ\cos\theta. That is one step short: divide the sine by it to get tan⁡θ=−512\tan\theta = -\frac{5}{12}.
  • C. Correct. The reference triangle is 5-12-13, so cos⁡θ=−1213\cos\theta = -\frac{12}{13} in quadrant II and tan⁡θ=513÷(−1213)=−512\tan\theta = \frac{5}{13} \div (-\frac{12}{13}) = -\frac{5}{12}.
  • D. Incorrect. 512\frac{5}{12} forgets the sign. In quadrant II the cosine is negative, so the tangent, sine over cosine, is negative.

Trap Right size, wrong sign

The reference angle gives the size of the value, never its sign. After you find it, ask which quadrant the angle is in. Watch for a choice that matches the size but has the wrong sign.

Trap Measuring to the y-axis

A reference angle is always measured to the x-axis. For 300∘300^\circ it is 360∘−300∘=60∘360^\circ - 300^\circ = 60^\circ, not 300∘−270∘=30∘300^\circ - 270^\circ = 30^\circ. Measuring to the y-axis swaps the sine and cosine values.

Calculator When a calculator isn’t faster

Exact values come from a reference angle and a sign, so hand work matches choices such as −32-\frac{\sqrt{3}}{2} directly. A calculator gives decimals such as −0.866-0.866, which you then have to recognize.

If you check with a calculator, set the angle mode to match the question, then compare decimals: 32≈0.866\frac{\sqrt{3}}{2} \approx 0.866, 22≈0.707\frac{\sqrt{2}}{2} \approx 0.707, and 3≈1.732\sqrt{3} \approx 1.732.