6.1 The Value of Sine, Cosine and Tangent for Angle θ, 0° < θ < 360°
On the Cartesian plane, a unit circle has radius \(1\text{ unit}\) and centre \((0,0)\). A point \(P\) moving anticlockwise from the positive \(x\)-axis forms an angle \(\theta\). The coordinate axes divide the circle into four quadrants. The signs of sine, cosine and tangent depend on the quadrant, while a corresponding reference angle \(\alpha\) helps determine their values. The main topic considers \(0^\circ < \theta < 360^\circ\); the axis values at \(0^\circ\) and \(360^\circ\) are also discussed as special cases.
Learning Objectives
- Make and verify conjectures about the values of sine, cosine, and tangent for angles in quadrants II, III, and IV with their corresponding reference angles.
- Determine the value of sine, cosine, and tangent for angles in quadrants II, III, and IV based on their corresponding reference angles.
- Determine the values of sine, cosine, and tangent for angles corresponding to special angles (\(30^\circ\), \(45^\circ\), and \(60^\circ\)) without using a scientific calculator.
- Determine angle \(\theta\) when sine, cosine or tangent is given, for \(0^\circ < \theta < 360^\circ\).
- Solve problems involving sine, cosine, and tangent.
The ASTC Quadrant Map and Unit Circle Coordinates
An all-in-one quick-reference visual showing positive signs in each quadrant, how to calculate reference angle \(\alpha\), and coordinates \((x, y) = (\cos \theta, \sin \theta)\).
Quadrant I
All (+)
\(\alpha = \theta\)
\((x>0,\ y>0)\)
Quadrant II
Sin (+)
\(\alpha = 180^\circ - \theta\)
\((x<0,\ y>0)\)
Quadrant III
Tan (+)
\(\alpha = \theta - 180^\circ\)
\((x<0,\ y<0)\)
Quadrant IV
Cos (+)
\(\alpha = 360^\circ - \theta\)
\((x>0,\ y<0)\)
Brief Explanation
What is a Unit Circle?
A unit circle is a circle with a radius of \(1\text{ unit}\) centred at \((0, 0)\). Angle \(\theta\) is measured from the positive \(x\text{-axis}\) in an anticlockwise direction.
What is a Reference Angle?
The corresponding reference angle, \(\alpha\), is always an acute angle (\(\alpha < 90^\circ\)) formed between the terminal line \(OP\) and the \(x\text{-axis}\).
Quadrant I: \(0^\circ < \theta < 90^\circ\)
\(\alpha = \theta\)
All positive: \(\sin \theta > 0\), \(\cos \theta > 0\), \(\tan \theta > 0\)
For \(a,b>0\), \(P=(a,b)\): \(\sin\theta=b\), \(\cos\theta=a\), \(\tan\theta=\dfrac{b}{a}\).
Quadrant II: \(90^\circ < \theta < 180^\circ\)
\(\alpha = 180^\circ - \theta\)
Only \(\sin \theta\) positive: \(\sin \theta > 0\), \(\cos \theta < 0\), \(\tan \theta < 0\)
For \(a,b>0\), \(P=(-a,b)\): \(\sin\theta=b\), \(\cos\theta=-a\), \(\tan\theta=-\dfrac{b}{a}\).
Quadrant III: \(180^\circ < \theta < 270^\circ\)
\(\alpha = \theta - 180^\circ\)
Only \(\tan \theta\) positive: \(\sin \theta < 0\), \(\cos \theta < 0\), \(\tan \theta > 0\)
For \(a,b>0\), \(P=(-a,-b)\): \(\sin\theta=-b\), \(\cos\theta=-a\), \(\tan\theta=\dfrac{b}{a}\).
Quadrant IV: \(270^\circ < \theta < 360^\circ\)
\(\alpha = 360^\circ - \theta\)
Only \(\cos \theta\) positive: \(\sin \theta < 0\), \(\tan \theta < 0\), \(\cos \theta > 0\)
For \(a,b>0\), \(P=(a,-b)\): \(\sin\theta=-b\), \(\cos\theta=a\), \(\tan\theta=-\dfrac{b}{a}\).
| Angle | \(\sin \theta\) | \(\cos \theta\) | \(\tan \theta\) |
|---|---|---|---|
| \(30^\circ\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{\sqrt{3}}\) |
| \(45^\circ\) | \(\dfrac{1}{\sqrt{2}}\) | \(\dfrac{1}{\sqrt{2}}\) | \(1\) |
| \(60^\circ\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | \(\sqrt{3}\) |
| Angle on an axis | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) |
|---|---|---|---|
| \(0^\circ\) | \(0\) | \(1\) | \(0\) |
| \(90^\circ\) | \(1\) | \(0\) | Undefined |
| \(180^\circ\) | \(0\) | \(-1\) | \(0\) |
| \(270^\circ\) | \(-1\) | \(0\) | Undefined |
| \(360^\circ\) | \(0\) | \(1\) | \(0\) |
Angles on the axes belong to no quadrant. Tangent is undefined where cosine is zero.
Quadrant II: \(\theta = 180^\circ - \alpha\)
Quadrant III: \(\theta = 180^\circ + \alpha\)
Quadrant IV: \(\theta = 360^\circ - \alpha\)
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Determine the quadrant and the corresponding reference angle \(\alpha\) for \(\theta = 239^\circ\).
Without using a calculator, find the value of \(\cos 150^\circ\).
On a unit circle, a point \(P\) at angle \(\theta\) has coordinates \((0.5592, -0.8290)\). State the values of \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\).
Common Mistakes
- Measuring the reference angle from the \(y\text{-axis}\). Students often calculate \(90^\circ - \theta\) or \(270^\circ - \theta\). Reference angle \(\alpha\) must always be measured relative to the horizontal \(x\text{-axis}\) (\(180^\circ\) or \(360^\circ\)).
- Inputting negative signs into the inverse trigonometric functions. When finding the reference angle \(\alpha\), using \(\sin^{-1}(-k)\) or \(\cos^{-1}(-k)\) directly in the calculator leads to confusion. Students should always use the positive magnitude \(|k|\) to find the acute reference angle \(\alpha\), then apply quadrant rules to find \(\theta\).
- Misidentifying positive/negative signs. Forgetting which trigonometric ratio is positive in each quadrant, for example assuming sine is positive in Quadrant III.
- Swapping \(x\) and \(y\) on the unit circle. Confusing \(\cos \theta\) with \(y\) and \(\sin \theta\) with \(x\). In a unit circle, \(x = \cos \theta\) and \(y = \sin \theta\).
Misconceptions
The reference angle \(\alpha\) can be obtuse or negative.
For angles strictly inside a quadrant, the reference angle is acute: \(0^\circ<\alpha<90^\circ\). Angles on the coordinate axes are evaluated directly using unit-circle coordinates.
Sine and cosine can have values greater than \(1\) or less than \(-1\).
Because \(x\) and \(y\) on a unit circle are bounded between \(-1\) and \(1\), the values of \(\sin \theta\) and \(\cos \theta\) must always lie within the range \([-1, 1]\).
Trigonometric ratios only exist for right-angled triangles with angles under \(90^\circ\).
Trigonometric functions extend to angles on the Cartesian plane and the unit circle, including angles beyond \(90^\circ\). Tangent is undefined whenever \(\cos\theta=0\).
Summary
- Unit Circle: Radius \(r=1\), centre \((0,0)\). A point on the unit circle has \((x,y)=(\cos\theta,\sin\theta)\), and \(\tan\theta=\dfrac{y}{x}\) only when \(x\ne0\).
- Quadrant Signs: Quadrant I: All \(+\); Quadrant II: Only \(\sin \theta\) \(+\); Quadrant III: Only \(\tan \theta\) \(+\); Quadrant IV: Only \(\cos \theta\) \(+\).
- Reference Angles (\(\alpha\)): QI: \(\alpha = \theta\); QII: \(\alpha = 180^\circ - \theta\); QIII: \(\alpha = \theta - 180^\circ\); QIV: \(\alpha = 360^\circ - \theta\).
- Special Angles: \(30^\circ\), \(45^\circ\), and \(60^\circ\) provide exact surd values for sine, cosine, and tangent.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 What is the corresponding reference angle, \(\alpha\), for \(\theta = 312^\circ\)?
2 In which quadrants is the value of \(\tan \theta\) negative?
B. Fill in the Blanks
3 For a point \(P(x, y)\) on a unit circle with angle \(\theta\), \(\cos \theta\) is equal to the .
4 The formula to calculate the corresponding reference angle \(\alpha\) for an angle \(\theta\) in Quadrant III is \(\alpha =\) .
C. Drag and Drop: Match Each Angle to Its Correct Trigonometric Value
Drag each expression to the correct value. On a phone, tap the expression first, then tap the value box.
