Trigonometric functions are commonly defined as ratios of two sides of a right-angled triangle containing the angle.

The tangent of an angle (tan(θ°)) is the y-coordinate of point Q of the triangle which extends from the unit circle.

The adjacent side of a right-angled triangle is the side which is next to both the right angle and the marked angle.

The opposite side of a right-angled triangle is the side which is opposite the marked angle.

Checking and securing understanding of tangent ratio problems

Maths

In this unit, pupils have explored the trigonometric ratios for right-angled triangles in a variety of situations. In the future unit, pupils explore what can be done when we have a triangle that is not right-angled.

In the prior unit, pupils were introduced to the trigonometric ratios and developed the skills to ascertain which ratio would be appropriate for a given problem. In this unit, pupils are introduced to a variety of situations where this knowledge is required, such as 3-D problems and elevation/depression.

Right-angled trigonometry

$$\tan(\theta)=\frac{\text{adj}}{\text{opp}}$$

$$\text{adj}\times\tan(\theta)=\text{opp}$$

$$\text{adj}=\frac{\tan(\theta)}{\text{opp}}$$

$$\theta=\arctan\left(\frac{\text{opp}}{\text{adj}}\right)$$

$$\text{adj}=\frac{\text{opp}}{\tan(\theta)}$$

For a right-angled triangle, the tangent ratio is $$\tan(\theta)=\frac{\text{opp}}{\text{adj}}$$, where $$\theta$$ is the angle in degrees. Which of these are equivalent forms?

Work out the length of $$x$$, to 1 decimal place.

Using the table of values, what is the value of $$\theta$$?

Work out the value of $$\theta$$, to the nearest degree.

Match the names of the sides, relative to the angle $$\theta$$.

The two triangles are similar. What is the length of $$x$$?

Triangle ABC and triangle DEF are similar. What is the length of $$x$$?

Work out the value of $$x$$ to 3 significant figures.

Work out the value of $$x$$ to 1 decimal place.

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