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Trigonometric functions for Edexcel A-level Maths

Trigonometric functions

This page covers the following topics:

1. Trigonometric functions
2. Exact values of trigonometric functions in radians
3. Secondary trigonometric functions

Sine, cosine and tangent are trigonometric ratios used to find angles and sides in a right-angled triangle. The sine of an angle can be calculated by dividing the opposite side to the angle by the hypotenuse. The cosine of an angle can be calculated by dividing the adjacent side to the angle by the hypotenuse. The tangent can be calculated by dividing the opposite side to the angle by the adjacent one. the three trigonometric ratios are usually given as sin, cos and tan. To find angles in a right-angled triangle, the inverse of these trigonometric ratios has to be used.

Trigonometric functions

The values for some specific trigonometric ratios are useful to know. The exact values of sine for the angles 0, 30ยฐ, 45ยฐ, 60ยฐ and 90ยฐ are are 0, 1/2, 1/โˆš2, โˆš3/2 and 1 respectively. The exact values of cosine for the angles 0, 30ยฐ, 45ยฐ, 60ยฐ and 90ยฐ are are 1, โˆš3/2, 1/โˆš2, 1/2 and 1 respectively. The exact values of tangent for the angles 0, 30ยฐ, 45ยฐ and 60ยฐ are 0, 1/โˆš3, 1 and โˆš3 respectively. The values for the specific trigonometric ratios can be converted into radians by dividing angles by 180ยฐ and multiplying them by ฯ€.

The values for multiples of these angles can be found using the method of quadrants. The value of the angle stays the same, however the sign changes depending on which quadrant the angle falls in. If it falls in the first quadrant, the value is positive for all three trigonometric ratios. If it falls in the second quadrant, only sine is positive. If it falls in the third quadrant, only tangent is positive. If it falls in the fourth quadrant, only cosine is positive.

Exact values of trigonometric functions in radians

The reciprocal trigonometric ratios are given by taking the reciprocal of the usual trigonometric ratios. Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent respectively. These are usually given as cosecant, sec and cot.

Inverse trigonometric ratios are used to calculate missing angles in right-angled triangles when the value of a trigonometric ratio is known. They are given as sinโปยน, cosโปยน and tanโปยน, and are calculated using the appropriate buttons on the calculator.

Different trigonometric ratios can be related to each other by some trigonometric identities. The reciprocal trigonometric ratios can be used in identities as follows: secยฒx = 1 + tanยฒx and cosecยฒx = 1 + cotยฒx.

Secondary trigonometric functions

1

Mary is cutting a piece of cardboard into a right-angle triangle to make a birthday card. Given that she wants an angle of 65ยฐ opposite a side of 4 cm, calculate the length of the adjacent side.

tanx = opposite รท adjacent
tan(65ยฐ) = 4 cm รท adjacent
adjacent = 4 cm รท tan(65ยฐ)
adjacent = 1.88 cm (to 3 s.f.)

1.88 cm

Mary is cutting a piece of cardboard into a right-angle triangle to make a birthday card. Given that she wants an angle of 65ยฐ opposite a side of 4 cm, calculate the length of the adjacent side.

2

Two right-angled triangles are arranged as shown in the diagram. Calculate the size of the angle x.

Consider the triangle on the left.
tan(30ยฐ) = opposite รท 3 cm
opposite = tan(30ยฐ) ร— 3 cm
opposite = 1.73 cm (to 3 s. f.)

1.73 cm โˆ’ 0.8 cm = 0.93 cm

Consider the triangle on the right.
sinx = opposite รท hypotenuse
sinx = 0.93 cm รท 7 cm
x = sinโปยน(0.93 cm รท 7 cm)
x = 7.65ยฐ (to 3 s. f.)

7.65ยฐ

Two right-angled triangles are arranged as shown in the diagram. Calculate the size of the angle x.

3

Jawaad wants to find the value of cos(11ฯ€/4). Explain how he would go about this.

cos(ฯ€/4) = 1/โˆš2
11ฯ€/4 lies in the second quadrant, therefore cosine is negative.
Therefore, cos(11ฯ€/4) = โˆ’1/โˆš2.

โˆ’1/โˆš2

Jawaad wants to find the value of cos(11ฯ€/4). Explain how he would go about this.

4

Find the value of x in the given diagram.

tanx = opposite รท adjacent
tanx = 4.5 cm รท 6 cm
x = tanโปยน(4.5 cm รท 6 cm)
x = 36.9ยฐ (to 3 s. f.)

36.9ยฐ

Find the value of x in the given diagram.

5

Rahul is building a toy boat, as shown in the diagram. Calculate the length of the pole attached to the boat he has to use to put the sail on his boat.

sinx = opposite รท hypotenuse
sin(50ยฐ) = opposite รท 8 cm
opposite = sin(50ยฐ) ร— 8 cm
opposite = 6.13 cm (to 3 s. f.)

length of stick = 6.13 cm + 1 cm = 7.13 cm

7.13 cm

Rahul is building a toy boat, as shown in the diagram. Calculate the length of the pole attached to the boat he has to use to put the sail on his boat.

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