0580 · 6.4Trigonometric functions

Unit circle and coordinates

0:00Two different angles can have exactly the same sine value. That means solving a trigonometric equation is not always finished when your calculator gives one angle. The graph can reveal another answer. In this video, we will connect turning around a circle with the shapes of sine, cosine and tangent graphs. You will learn to recognise, sketch and interpret these graphs over one complete turn. We will then use their shapes to find every solution to an equation.

0:24Our interval runs from zero degrees to three hundred and sixty degrees, including both endpoints. Every answer must belong to that stated interval. You already know sine, cosine and tangent as ratios in right-angled triangles. Sine compares the opposite side with the hypotenuse, while cosine compares the adjacent side with the hypotenuse. Tangent compares the opposite side with the adjacent side. Those ideas were developed in the video on right-angled triangles.

0:51A triangle on its own does not explain angles larger than ninety degrees. To understand those angles, imagine a radius turning around a circle. We measure its angle anticlockwise from the horizontal direction pointing right. A complete turn brings the radius back to its starting position at three hundred and sixty degrees. Choose a circle with radius one, called a unit circle. For an angle in the upper-right region, dropping a perpendicular creates a right-angled triangle.

1:16Its hypotenuse has length one, so the horizontal coordinate equals cosine x. The vertical coordinate equals sine x because dividing its height by one leaves the same value. We extend that interpretation all the way around the circle. Cosine tells us the point’s horizontal coordinate, and sine tells us its vertical coordinate. Left of the centre, the horizontal coordinate is negative, and below it, so is the vertical one. This explains negative trigonometric values without pretending that a triangle has negative side lengths.

1:47At zero degrees, the point is at the far right of the circle. Its horizontal coordinate is one and its height is zero. After a turn of ninety degrees, it reaches the top. Cosine is then zero, while sine is one, because the point is directly above the centre. At one hundred and eighty degrees, the point is at the far left. Its cosine is negative one and its sine is zero.

2:09At two hundred and seventy degrees, it reaches the bottom, giving cosine zero and sine negative one. Completing the turn restores the starting values at three hundred and sixty degrees.

Sine and cosine graphs

2:19Plotting the vertical coordinate against the angle gives the sine graph. It begins at zero, rises smoothly to one, and returns to zero halfway through the turn. It then falls to negative one before returning to zero. These five key positions give the structure of a reliable sketch. Join the key points with a smooth curve, not straight segments. The maximum value is one, and the minimum is negative one.

2:43Sine is positive between zero degrees and one hundred and eighty degrees because the moving point is above the centre. It is negative during the lower half of the circle. The cosine graph records the horizontal coordinate instead. It starts at one, reaches zero at ninety degrees, and falls to negative one at one hundred and eighty degrees. It returns through zero at two hundred and seventy degrees. The final value is one because the point has returned to the rightmost position.

3:10Cosine is positive where the circle lies to the right of its centre. That includes angles below ninety degrees and above two hundred and seventy degrees. Between ninety degrees and two hundred and seventy degrees, cosine is negative. Reading the circle explains the signs instead of leaving them as facts to memorise. Both graphs repeat after three hundred and sixty degrees because the point completes one full turn. The angle needed for a pattern to repeat is called its period.

3:35Both sine and cosine have a period of three hundred and sixty degrees. Their values never leave the interval from negative one to one. That restriction immediately helps us interpret equations. Sine x = 2 has no solution because the sine graph never reaches that height. Likewise, cosine cannot equal negative three. Before calculating any angle, check whether the requested value is possible for the function.

Tangent graph and asymptotes

4:00Tangent has a different shape because it compares the vertical coordinate with the horizontal coordinate. Tangent x equals sine x divided by cosine x wherever cosine is not zero. This matches opposite divided by adjacent in a right-angled triangle. Unlike sine and cosine, tangent is not limited to values between negative one and one. As the angle approaches ninety degrees from below, the horizontal coordinate becomes very small and positive.

4:26The vertical coordinate remains close to one. Dividing by a smaller positive number makes tangent increasingly large. At ninety degrees itself, cosine is zero, so tangent would require division by zero and is undefined. Just beyond ninety degrees, the horizontal coordinate is small and negative. Tangent therefore has a large negative value on that side. The graph does not connect across ninety degrees. The same problem occurs at two hundred and seventy degrees, where cosine is also zero.

4:55The dashed vertical lines are called asymptotes. They mark positions that the branches approach without meeting. Draw the tangent graph in three separate pieces within our interval, with each one rising from left to right. It crosses the horizontal axis at zero degrees, one hundred and eighty degrees, and three hundred and sixty degrees. A half-turn changes the signs of both the horizontal and vertical coordinates. Dividing two negative coordinates gives the same ratio as dividing their positive counterparts.

5:23Tangent therefore repeats after one hundred and eighty degrees. This shorter period explains why tangent solutions are commonly separated by one hundred and eighty degrees.

Solving sine equations

5:31An equation such as sine x = ½ asks where the sine graph has that height. Drawing a horizontal line makes the solutions visible as intersection points. A calculator usually supplies just one angle, not every intersection. We need the graph’s shape or the circle’s symmetry to find the others. A useful starting point is a reference angle. This is the acute angle associated with the required trigonometric ratio. We then locate angles with that ratio’s correct sign in our stated interval.

6:00For exact values, use the knowledge from the video on exact trigonometric values rather than relying on decimal approximations. Our first worked example is sine x = √3/2. We recognise that the sine of sixty degrees equals √3/2. The reference angle is therefore sixty degrees. Since the required value is positive, we look in the upper half of the circle. The first solution is sixty degrees. The other point at the same height is its reflection across the vertical axis.

6:29Its angle is one hundred and eighty degrees minus sixty degrees. Subtracting gives one hundred and twenty degrees, so the complete answer is sixty degrees or one hundred and twenty degrees. There are no further solutions in the stated interval because sine is negative in the lower half of the circle. Neither endpoint works, since sine is zero there. Both answers are exact and need no rounding. This method must work without a calculator on Paper two, the non-calculator paper.

Cosine and tangent examples

6:57Our second example is two cosine x + 1 = 0. Subtract one from both sides to obtain two cosine x = −1. Dividing both sides by two gives cosine x = −½. Isolating the trigonometric function first reveals the value that our graph must reach. The cosine of sixty degrees is one half, so the reference angle is sixty degrees. We need negative cosine, which means points to the left of the circle’s centre.

7:24The upper-left solution is one hundred and eighty degrees minus sixty degrees. This gives one hundred and twenty degrees. The lower-left solution is one hundred and eighty degrees plus sixty degrees. Adding gives two hundred and forty degrees. Both angles lie in the stated interval and have cosine negative one half. Substituting that value gives −1 + 1, which equals zero, so both satisfy the original equation. The symmetry used for sine and cosine is different.

7:53Equal sine values mean equal vertical coordinates, so we reflected across the vertical axis. Equal cosine values mean equal horizontal coordinates, so we reflect across the horizontal axis. Using the same angle rule indiscriminately for both functions can put an answer in the wrong region. For a calculator example, solve tangent x = −1.5. Put the calculator in degree mode because our interval is measured in degrees. The inverse tangent of positive one point five gives a reference angle of about fifty-six point three degrees.

8:25Keep the full calculator value for the next steps. Tangent is negative when the horizontal and vertical coordinates have opposite signs. That happens in the upper-left and lower-right regions. For the first solution, subtract the reference angle from one hundred and eighty degrees. The result is approximately one hundred and twenty-three point six nine degrees. Add one hundred and eighty degrees to obtain the next solution because tangent repeats after a half-turn.

8:50This gives approximately three hundred and three point six nine degrees. Round the final angles to one decimal place. Our answers are one hundred and twenty-three point seven degrees and three hundred and three point seven degrees. This is a Paper four calculator method. Unless the question specifies otherwise, give calculated angles to one decimal place and other numerical answers to three significant figures. Do not round the reference angle halfway through the working.

9:15Early rounding can change the last digit of the final answer.

Exam skills and recap

9:20Endpoints need particular care, as this short example shows. To solve sine x = 0, read where the sine graph meets the horizontal axis. The intersections give zero degrees, one hundred and eighty degrees, and three hundred and sixty degrees. The first and last angles describe the same circle position, but both belong to the interval. In an examination, recognise means identify the function from its characteristic features. A wave starting at zero suggests sine, while one that starts at its maximum suggests cosine.

9:50Separate increasing branches with vertical asymptotes identify tangent. Check the axis labels and scale rather than identifying a graph from its appearance alone. For a sketch, show the correct shape and label the important values. A sine sketch should show its zeros, maximum and minimum in their correct positions. Cosine needs its different starting value and corresponding turning points. Tangent requires separate branches, zeros and clearly positioned asymptotes, never a continuous line across undefined angles.

10:20For an equation, show the algebra that isolates the function before finding angles. State the reference angle or calculator result, then show how you obtained every solution. Missing the second intersection is a common mistake because an inverse function gives only one principal value. Counting graph intersections provides a useful check on your final answer. Also distinguish an inverse function from a reciprocal. The inverse sine of a number gives an angle with that sine.

10:47It does not mean one divided by the sine. A negative trigonometric value does not automatically require a negative angle either, as our positive-angle cosine and tangent solutions demonstrated. The central connection is that sine records vertical position, cosine records horizontal position, and tangent records their ratio. The circle explains their signs, their graph shapes and their repeating patterns. Solving an equation means locating every angle where the function reaches the required value.

11:14Use the interval, symmetry and graph together to make your answers complete.