Exact Trig Values

Recall the exact-value table.

Recall the cosine row of the exact-value table.

Think about the two shorter sides of a right-angled isosceles triangle.

Use the exact-value table: the 90° column for sine and the 0° column for cosine. Sine gets bigger from 0° to 90°, and cosine gets smaller.

Use the exact-value table: the 0° column for sine and tan, and the 90° column for cosine. Picture a right-angled triangle with a very small angle: how long is the side opposite it?

Use the 45° entry, not a rounded decimal.

1. \sin45°=\dfrac{\sqrt2}{2}.
2. The equivalent form \dfrac1{\sqrt2} is also exact.

Recall the cosine row: the value is greater than one half.

1. \cos30°=\dfrac{\sqrt3}{2}.

Use the 60° sine entry.

1. \sin60°=\dfrac{\sqrt3}{2}.

Sine and cosine have the same value at 45°.

1. \cos45°=\dfrac{\sqrt2}{2}=\dfrac1{\sqrt2}.

At 30° and 60°, the tangent values are reciprocals.

Part (a)

1. \tan30°=\dfrac1{\sqrt3}=\dfrac{\sqrt3}{3}.

Part (b)

1. \tan60°=\sqrt3.

Recall the three values and compare their sizes.

1. \sin30°=\dfrac12, \tan30°=\dfrac1{\sqrt3} and \cos30°=\dfrac{\sqrt3}{2}.
2. These are approximately 0.5, 0.577 and 0.866, respectively.
3. So the order is \sin30°, then \tan30°, then \cos30°.

Substitute each exact value before adding.

1. \sin30°+\cos60°=\dfrac12+\dfrac12=1.

Replace each trig value with its exact value, then work out the expression.

1. \cos60°=\dfrac12 and \tan45°=1.
2. 6\times\dfrac12+1=3+1=4.

Use the ratio linking x and the hypotenuse.

1. \sin30°=\dfrac{x}{18}=\dfrac12.
2. x=18\times\dfrac12=9 cm.

The 7 cm side is next to the 60° angle and x is the hypotenuse. Which ratio links these two sides?

1. The 7 cm side is adjacent to the 60° angle and x is the hypotenuse, so use cosine.
2. \cos60°=\dfrac{7}{x} and \cos60°=\dfrac12, so \dfrac{7}{x}=\dfrac12.
3. x=7\times2=14 cm.

Find the length of the vertical side first. Which exact value links the two shorter sides when the angle is 45°?

1. The vertical side is opposite the 45° angle and the 7 cm side is adjacent to it, so use tangent.
2. \tan45°=\dfrac{\text{vertical side}}{7} and \tan45°=1, so the vertical side is 7 cm.
3. Area =\dfrac12\times7\times7=24.5 cm².

Replace each trig value with its exact value first. Square the whole of \cos60° before multiplying by 4.

1. \cos60°=\dfrac12, \tan45°=1 and \sin0°=0.
2. 4\times\left(\dfrac12\right)^2+2\times1-0=4\times\dfrac14+2=1+2=3.

Use the 45° angle to find the height of the rectangle first.

1. The height is opposite the 45° angle and the 10 cm side is adjacent to it.
2. Height =10\times\tan45°=10\times1=10 cm.
3. Perimeter =10+10+10+10=40 cm.

In each triangle, decide whether the length you want is opposite or next to the given angle, then use the exact value.

Part (a)

1. The pole is opposite the 30° angle and the 8 m rope is the hypotenuse, so use sine.
2. Height =8\times\sin30°=8\times\dfrac12=4 m.

Part (b)

1. The distance along the ground is next to the 60° angle and the 6 m rope is the hypotenuse, so use cosine.
2. Distance =6\times\cos60°=6\times\dfrac12=3 m.

Write \sin30° as a fraction using the sides of this triangle, then rearrange.

Part (a)

1. x is opposite the right angle, so it is the hypotenuse, the longest side of the triangle.
2. 2.5 cm is shorter than the 5 cm side, so it cannot be right.

Part (b)

1. The 5 cm side is opposite the 30° angle and x is the hypotenuse, so \sin30°=\dfrac{5}{x}.
2. \dfrac12=\dfrac{5}{x}, so x=5\times2=10 cm.