Sine and Cosine Function Values
Introduced| English | Meaning |
|---|---|
| exact value/eɡˈzækt ˈvæljuː/ | A value stated without rounding or approximation. |
| unit circle/ˈjuːnɪt ˈsɜːkl/ | A circle with radius one, usually centred at the origin. |
| cosine/ˈkəʊsaɪn/ | For an acute angle in a right-angled triangle, the adjacent side divided by the hypotenuse. |
| sine/saɪn/ | For an acute angle in a right-angled triangle, the opposite side divided by the hypotenuse. |
| radians/ˈreɪdɪənz/ | Units of angle defined by arc length divided by radius; one full turn equals two pi of these units. |
| isosceles right triangle | isosceles right triangle |
| equilateral triangle/ˌiːkwɪˈlætərəl ˈtraɪæŋɡl/ | A triangle with all sides equal and all angles 60°, source of the exact 30° and 60° trig values. |
| reference angle/ˈrefrəns ˈæŋɡl/ | The acute angle to the horizontal axis. |
| quadrant/ˈkwɒdrənt/ | One of the four regions of the coordinate plane, determining the signs of trig values. |
| period/ˈpɪərɪəd/ | The time taken for one complete oscillation or cycle. |
Keep values exact before measuring
- A wheel seat travels 4 m from its centre. At $150^\circ$ from the positive horizontal direction, its height above the centre is exactly 2 m. An exact value 精确值 is written without rounding, using expressions such as fractions or radicals.
- On the unit circle 单位圆, cosine 余弦 is the horizontal coordinate and sine 正弦 is the vertical coordinate. Special triangles give these values without a calculator: $\sqrt2/2$ is exact, while 0.707 is a rounded approximation.
Anchor the values on the axes
- In radians 弧度, a full turn is $2\pi$. At $0$ and $\pi/2$ ($0^\circ$ and $90^\circ$), the unit-circle points are $(1,0)$ and $(0,1)$. Read cosine first, then sine.
- At $\pi$ and $3\pi/2$ ($180^\circ$ and $270^\circ$), the points are $(-1,0)$ and $(0,-1)$. These axis points need no acute triangle: for example, $\cos\pi=-1$ and $\sin(3\pi/2)=-1$.
What is $\sin 0$?
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The unit-circle point is(1,0); sine is the vertical coordinate,0.
What is $\cos 0$?
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The unit-circle point is(1,0); cosine is the horizontal coordinate,1.
What is $\cos\pi$?
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The half-turn point is(−1,0), so cosine is−1.
Derive the 45-degree values
- An isosceles right triangle 等腰直角三角形 has equal legs. If each leg is 1, the hypotenuse is $\sqrt{1^2+1^2}=\sqrt2$, and each acute angle is $45^\circ$.
- Divide each leg by the hypotenuse to reach the unit circle. Multiplying the top and bottom of $1/\sqrt2$ by $\sqrt2$ gives $\sqrt2/2$. Thus $\cos(\pi/4)=\sin(\pi/4)=\sqrt2/2$. Both are about 0.707, but their exact form is the radical expression.
Which exact pair follows from a right triangle with equal legs 1 and hypotenuse $\sqrt 2$?
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Each coordinate ratio is 1/√2=√2/2.
Derive the 30- and 60-degree values
- Split an equilateral triangle 等边三角形 of side 2 with an altitude. Its $60^\circ$ angle is halved to $30^\circ$, and the opposite side is halved to 1. The resulting right triangle has hypotenuse 2 and long leg $\sqrt{2^2-1^2}=\sqrt3$.
- Therefore $\sin30^\circ=1/2$ and $\cos30^\circ=\sqrt3/2$. At $60^\circ$, opposite and adjacent legs swap: $\sin60^\circ=\sqrt3/2$ and $\cos60^\circ=1/2$. In radians these angles are $\pi/6$ and $\pi/3$.
What is $\sin 30^\circ$?
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Half an equilateral triangle gives opposite 1 and hypotenuse 2, so sine=1/2.
Find the reference angle
- For a non-axis terminal ray, its reference angle 参考角 is the acute angle to the horizontal axis. It gives the magnitudes of the coordinates; the quadrant 象限 supplies their signs. In quadrant I the reference angle is the original angle.
- For angles between $0^\circ$ and $360^\circ$, use $180^\circ-\theta$ in quadrant II, $\theta-180^\circ$ in III and $360^\circ-\theta$ in IV. Thus $150^\circ$, $210^\circ$ and $330^\circ$ all have reference angle $30^\circ$. Axis points are read directly.
What is the reference angle of 150 degrees, in degrees?
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Quadrant II uses 180−150=30 degrees.
Apply signs to exact magnitudes
- At $150^\circ$, the reference angle is $30^\circ$. Quadrant II has negative horizontal and positive vertical coordinates, so $\cos150^\circ=-\sqrt3/2$ and $\sin150^\circ=1/2$.
- At $225^\circ$, both coordinates are negative: $(\cos\theta,\sin\theta)=(-\sqrt2/2,-\sqrt2/2)$. At $330^\circ$, the point is $(\sqrt3/2,-1/2)$. Find the reference angle first, then assign each coordinate's sign.
What is $\sin 150^\circ$?
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The 30-degree reference angle gives magnitude 1/2, and sine is positive in quadrant II.
What is the exact value of $\cos 225^\circ$?
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The reference angle is 45 degrees. Quadrant III makes the horizontal coordinate negative.
What is $\sin 330^\circ$?
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The reference angle is 30 degrees and sine is negative in quadrant IV.
Read the unit before finding a value
- Convert a radian input to a familiar direction when useful: $5\pi/6$ radians is $150^\circ$, so its sine is $1/2$ and its cosine is $-\sqrt3/2$. The radian number is an input angle, not a sine or cosine output.
- At $-\pi/4$ radians, turn clockwise by $45^\circ$. The point is $(\sqrt2/2,-\sqrt2/2)$, so cosine is positive and sine is negative. A negative angle does not automatically make both function values negative.
Reduce turns and use reflection
- The fundamental period 周期 of sine and cosine is $2\pi$ radians, or $360^\circ$. Adding an integer multiple of a full turn preserves both values: $\sin(13\pi/6)=\sin(\pi/6)=1/2$. A half-turn generally changes both signs instead of preserving the values.
- Reflection across the horizontal axis gives $\cos(-\theta)=\cos\theta$ and $\sin(-\theta)=-\sin\theta$. For example, at $-\pi/6$ the point is $(\sqrt3/2,-1/2)$. Whole-turn reduction and reflection work with exact expressions, without decimal conversion.
The fundamental period of sine and cosine is ____ radians.
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One full turn is 2π radians, equivalent to 360 degrees. The number 360 is not the period in radians.
What is $\sin(13\pi/6)$?
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Subtract 2π to obtainπ/6; a full turn preserves the value 1/2.
For every real angle, $\cos(-\theta)=\cos\theta$.
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Reflection across the horizontal axis preserves the horizontal coordinate, so cosine is unchanged.
Scale coordinates without changing ratios
- On a radius-$r$ circle, $x=r\cos\theta$ and $y=r\sin\theta$. For $r=4$ and $\theta=150^\circ$, the point is $(-2\sqrt3,2)$, while cosine and sine remain $-\sqrt3/2$ and $1/2$.
- Coordinates can be larger than 1 when the radius is larger than 1. The bounds $-1\le\sin\theta\le1$ and $-1\le\cos\theta\le1$ apply to the normalised ratios, not to every coordinate on every circle.
Keep special-angle values exact
Use a reference angle and quadrant signs. Compare exact function values with coordinates on a larger circle.
A radius-4 circle has angle 150 degrees. What is its vertical coordinate?
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The coordinate is r×sinθ=4×1/2=2. The sine value itself is 1/2.
Recover a value and choose its sign
- Suppose $\sin\theta=-3/5$ and $\theta$ is in quadrant III. The identity gives $\cos^2\theta=1-9/25=16/25$, so the possible cosine values are $\pm4/5$.
- Quadrant III selects $\cos\theta=-4/5$. Squaring loses the sign, so the identity alone would not choose between the two roots. A sine value also need not identify a unique angle; retain the stated quadrant or angle interval.
Sine is−3/5 and the angle is in quadrant III. What is cosine?
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The identity gives cosine magnitude 4/5; quadrant III selects−4/5=−0.8.
Distinguish a bound from a transformed output
- An equation $\sin\theta=1.4$ has no real solution. In a measurement problem that result may indicate an error; in an equation it can simply mean the target is unreachable. The same bound applies to cosine.
- A transformed output can exceed 1: $h(t)=5+3\sin t$ has minimum 2 and maximum 8. The sine inside still lies in $[-1,1]$. State whether a number is a function value, an angle or a scaled/shifted output before applying a bound.
Derive the special magnitudes from triangles, use the reference angle and quadrant for signs, and reduce full turns in the stated units. Keep radicals and fractions exact until rounding is requested; circle scaling changes coordinates rather than the sine and cosine ratios.
For a real angle, the value of sine always lies in which interval?
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The unit-circle vertical coordinate lies in[−1,1]. Larger-radius coordinates and transformed outputs can have a different range.
Select all true statements about the untransformed sine and cosine functions.
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For real inputs, their outputs lie in[−1,1], cosine of 0 is 1 and the period is 2π radians. Sine itself cannot equal 2.
What is the range of $h(t)=5+3\sin t$ for all real t?
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Multiplying[−1,1] by 3 gives[−3,3]; adding 5 gives[2,8].
Explain values from coordinates and symmetry
- Find $\sin(\pi/2)$ and $\cos(\pi/2)$ from the unit-circle point. Explain why a full turn added to the angle leaves both values unchanged.
- Then find $\sin(13\pi/6)$ and $\cos(7\pi/6)$ exactly. State the reduced or reference angle and the sign in each case.
Find sine and cosine at $\pi/2$, explaining the coordinates and a full-turn repeat. Then find $\sin(13\pi/6)$ and $\cos(7\pi/6)$ exactly with their signs.
Use the axis point for the quarter turn and a 30-degree reference angle for the other two values.
What is $\sin(\pi/2)$?
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The quarter-turn point is(0,1), so sine is 1. The other choices are distinct axis values.