Sine, Cosine, and Tangent
Introduced| English | Meaning |
|---|---|
| sine/saɪn/ | For an acute angle in a right-angled triangle, the opposite side divided by the hypotenuse. |
| cosine/ˈkəʊsaɪn/ | For an acute angle in a right-angled triangle, the adjacent side divided by the hypotenuse. |
| tangent/ˈtændʒənt/ | The ratio of sine to cosine of an angle, or opposite side to adjacent side in a right-angled triangle. |
| standard position/ˈstændəd pəˈzɪʃn/ | An angle drawn from the positive x-axis with its vertex at the origin. |
| terminal ray | terminal ray |
| unit circle/ˈjuːnɪt ˈsɜːkl/ | A circle with radius one, usually centred at the origin. |
| quadrant/ˈkwɒdrənt/ | One of the four regions of the coordinate plane, determining the signs of trig values. |
| radians/ˈreɪdɪənz/ | Units of angle defined by arc length divided by radius; one full turn equals two pi of these units. |
Locate a point on a rotating wheel
- Measure a wheel seat's position from the wheel's centre, with horizontal coordinate $x$ and vertical coordinate $y$. A seat at $(3,4)$ on a wheel of radius 5 satisfies $3^2+4^2=5^2$.
- If the wheel doubles in size at the same angle, the coordinates double to $(6,8)$, but the ratios $x/r$ and $y/r$ stay the same. Ratios describe the direction independently of the wheel's size.
Start with an acute triangle angle
- In a right triangle, identify the opposite side and adjacent side relative to the chosen acute angle. The hypotenuse is opposite the right angle. Sine 正弦 is opposite divided by hypotenuse; cosine 余弦 is adjacent divided by hypotenuse.
- Tangent 正切 is opposite divided by adjacent. For sides opposite 5, adjacent 12 and hypotenuse 13, the ratios are $\sin\theta=5/13$, $\cos\theta=12/13$ and $\tan\theta=5/12$. Similar triangles give the same ratios at the same acute angle.
For a chosen acute angle in a right triangle, $\sin\theta$ equals…
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Sine uses the opposite side divided by the hypotenuse. The chosen angle determines which leg is opposite.
State how an angle is measured
- An angle in standard position 标准位置 starts at the positive horizontal axis with its vertex at the origin. Its terminal ray 终边 shows the final direction. Counterclockwise turns are positive; clockwise turns are negative.
- More than one turn is allowed. Angles $90^\circ$, $450^\circ$ and $-270^\circ$ end in the same direction. Their turn counts differ, but their circle coordinates at a fixed radius are the same.
Which angle has the same terminal direction as 450 degrees?
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Subtract one full 360-degree turn:450−360=90 degrees.
Define the functions on any circle
- On a circle centred at the origin with radius $r>0$, a terminal point $(x,y)$ satisfies $x^2+y^2=r^2$. Define $\cos\theta=x/r$ and $\sin\theta=y/r$. These use signed coordinates, not just positive side lengths.
- When $x\ne0$, $\tan\theta=y/x$. For the point $(-3,4)$ with $r=5$, the values are $\cos\theta=-3/5$, $\sin\theta=4/5$ and $\tan\theta=-4/3$. A radius of 5 does not make the cosine equal to the raw coordinate $-3$.
A radius-5 circle has terminal point(-3,4). What is cosine?
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Cosine=x/r=−3/5=−0.6.
At terminal point(-3,4), give tangent to three decimal places.
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Tangent=y/x=4/(−3)=−4/3≈−1.333.
Normalise to the unit circle
- The unit circle 单位圆 has radius 1. Its point is $(\cos\theta,\sin\theta)$; on a circle of radius $r$, the point is $(r\cos\theta,r\sin\theta)$. Dividing both coordinates by $r$ gives the corresponding unit-circle point.
- The point $(12,5)$ on a radius-13 circle therefore corresponds to $(12/13,5/13)$ on the unit circle. Check the invariant $\cos^2\theta+\sin^2\theta=1$, which follows from dividing $x^2+y^2=r^2$ by $r^2$.
On the unit circle, a terminal point at angle $\theta$ has coordinates…
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Radius 1 makes cosine equal to the horizontal coordinate and sine equal to the vertical coordinate. On radius r, multiply both values by r.
Read signs from the quadrant
- A quadrant 象限 is one of the four regions between the coordinate axes. In quadrant I, $x,y>0$, so all three ratios are positive. In quadrant II, $x<0,y>0$, so cosine and tangent are negative while sine is positive.
- In quadrant III, $x,y<0$, so sine and cosine are negative but tangent is positive. In quadrant IV, $x>0,y<0$, so cosine is positive while sine and tangent are negative. Points on the axes are not inside a quadrant; read their values directly.
Which signs occur inside quadrant III?
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Both coordinates are negative. Dividing either by the positive radius keeps it negative; their quotient is positive.
Handle points on the axes
- At $0^\circ$, the unit-circle point is $(1,0)$; at $180^\circ$, it is $(-1,0)$. Sine and tangent are zero at both angles, while cosine is 1 and $-1$ respectively.
- At $90^\circ$ and $270^\circ$, the points are $(0,1)$ and $(0,-1)$. Cosine is zero, so tangent is undefined: division by zero does not give a real number. Sine is 1 and $-1$ respectively.
At 90 degrees, tangent is a real number equal to infinity.
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The horizontal coordinate is zero. Division by zero is undefined; infinity is not a real tangent value.
Connect radians to arc length
- Radians 弧度 measure an angle by arc length $s$ divided by radius $r$. For a positive counterclockwise turn, $\theta=s/r$. One radian cuts off an arc equal to the radius. Conversely, $s=r\theta$. For $r=6$ cm and $\theta=\pi/3$ radians, $s=6(\pi/3)=2\pi$ cm.
- A full turn has arc length $2\pi r$, hence angle $2\pi$ radians. For an arc of 10 cm on a radius-5 cm circle, $\theta=10/5=2$ radians. A clockwise angle is negative; an ordinary arc length remains nonnegative, so use the angle's magnitude for that length.
One full counterclockwise turn has angle $2\pi$ radians.
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Circumference divided by radius is 2πr/r=2π; the same full turn is 360 degrees.
A positive counterclockwise arc is 10 cm long on a radius-5 cm circle. What is its angle in radians?
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Angle=arc length/radius=10/5=2 radians.
A circle has radius 6 cm. A positive angle is $\pi/3$ radians. Find its arc length in centimetres to three decimal places.
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Arc length = radius × angle in radians = 6×π/3 = 2π ≈ 6.283 cm.
Convert before using a calculator
- Since $180^\circ=\pi$ radians, multiply degrees by $\pi/180$ to obtain radians, or radians by $180/\pi$ to obtain degrees. Thus $150^\circ=5\pi/6$ radians and $-\pi/3$ radians equals $-60^\circ$.
- The input unit matters: $\sin(90^\circ)=1$ exactly and $\sin(\pi/2)=1$ in radian notation. Entering 90 in radian mode gives a different value. Match the calculator mode to the stated angle, and round only the final calculation.
What is 150 degrees in radians?
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Multiply 150 byπ/180 to obtain 5π/6.
What is $-\pi/3$ radians in degrees?
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Multiply−π/3 by 180/π to obtain−60 degrees.
Interpret tangent and repeated angles
- Tangent is the slope of the terminal line through the origin: $\tan\theta=y/x=\sin\theta/\cos\theta$, where $x\ne0$. It is undefined at $\theta=\pi/2+k\pi$ radians for integer $k$. Sine and cosine are defined for every real angle and lie between $-1$ and 1; tangent is not restricted to that range.
- Adding a full turn leaves sine and cosine unchanged. Tangent also repeats after a half-turn, since both $x$ and $y$ change sign and their ratio stays the same. A function value alone does not select a unique angle; recovering an angle later requires a stated range or other information.
Keep the ratios while changing the circle
Compare signed coordinates, radius-normalised values and tangent. Include axis points and multiple or clockwise turns.
Where cosine is nonzero, tangent equals sine divided by ____.
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Tangent is sine divided by cosine. If cosine is zero, the ratio is undefined.
Select all true statements with their stated conditions.
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The first three statements include the conditions needed for their ratios. A full turn is 360 degrees, or 2π radians.
Knowing only a sine value always identifies one unique real angle.
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A full turn can be added without changing sine. Other angles within a turn can share the same sine too.
Recover a coordinate with the correct sign
- Suppose $r=13$, $x=-5$, and the terminal point is in quadrant III. The circle equation gives $y^2=13^2-(-5)^2=144$. The quadrant selects $y=-12$, rather than $+12$.
- Therefore $\cos\theta=-5/13$, $\sin\theta=-12/13$ and $\tan\theta=12/5$. A square-root calculation gives the magnitude; the location supplies its sign. Without a quadrant or another restriction, the sign of $y$ would remain uncertain.
State the angle units and direction. Use $x/r$ for cosine, $y/r$ for sine and $y/x$ for tangent when $x\ne0$. Coordinates supply signs, scaling by the radius preserves the ratios, and repeated angles explain why a value does not identify one angle by itself.
On a radius-13 circle, x=−5 and the point is in quadrant III. What is y?
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The circle equation gives y²=169−25=144. Quadrant III selects the negative root,−12.
Explain the three ratios and an exception
- For an acute angle in a right triangle with opposite side 5, adjacent side 12 and hypotenuse 13, find sine, cosine and tangent. State the corresponding point on the unit circle.
- Then consider a radius-13 circle at point $(0,13)$. Find sine and cosine, and explain why tangent has no real value there.
For an acute right-triangle angle with opposite 5, adjacent 12 and hypotenuse 13, find all three ratios and its unit-circle point. Then explain the values at the radius-13 point $(0,13)$.
Normalise coordinates by the radius, and check the denominator before evaluating tangent.
For an acute right-triangle angle with opposite side 5, adjacent side 12 and hypotenuse 13, what is its sine?
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Sine is opposite divided by hypotenuse:5/13.