In 4th ESO you already extended sine and cosine beyond the right-angled triangle, using a circle of radius $1$. In Bachillerato this idea is formalised with radians and is used constantly to calculate exact values, solve triangles and model phenomena that rotate or repeat. Review the key idea and add the tools you are missing: the special angles and careful handling of signs.
In a right-angled triangle with an acute angle $\alpha$, the sine is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the adjacent side to the hypotenuse. If that triangle is placed with the vertex of the angle at the origin and the hypotenuse as the radius of a circle of radius $1$, the sine and cosine are, directly, the coordinates of the point where the arm of the angle meets the circle.
Practice★☆☆Not started
Calculate the exact value of the sine, cosine and tangent of the smallest angle of a right-angled triangle, knowing that its shorter sides measure $5$ cm and $12$ cm.
Calculate the exact value of the sine, cosine and tangent of the smallest angle of a right-angled triangle, knowing that its shorter sides measure $5$ cm and $12$ cm.
There are three angles whose sines, cosines and tangents can be calculated exactly from two very particular triangles: the isosceles right-angled triangle (for $45º$) and half of an equilateral triangle (for $30º$ and $60º$). That is why their values are fractions with roots, not rounded decimals.
Practice★☆☆Not started
Convert the angles of $135^\circ$ and $330^\circ$ to radians, and express the angles of $\frac{5\pi}{6}$ rad and $\frac{7\pi}{4}$ rad in sexagesimal degrees.
Convert the angles of $135^\circ$ and $330^\circ$ to radians, and express the angles of $\frac{5\pi}{6}$ rad and $\frac{7\pi}{4}$ rad in sexagesimal degrees.
The Pythagorean identity and the sign in each quadrant
Since $P(\cos\alpha, \sin\alpha)$ is always at distance $1$ from the origin (the radius of the unit circle), Pythagoras directly gives the most widely used relationship in the whole of trigonometry. But when you solve for a ratio using a square root, the sign is not decided by the root: it is decided by the quadrant in which the angle lies.
Practice★☆☆Not started
Knowing that $\cos \alpha = \frac{4}{5}$ and that $\alpha$ is an acute angle, find the exact value of $\sin \alpha$ and $\tan \alpha$ using fundamental trigonometric…
Knowing that $\cos \alpha = \frac{4}{5}$ and that $\alpha$ is an acute angle, find the exact value of $\sin \alpha$ and $\tan \alpha$ using fundamental trigonometric identities.
Given that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.
Given that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.
To find the sign or the value of the ratios of a large angle, first reduce it to $[0º,360º)$ by subtracting complete revolutions, and then locate its quadrant.
Practice★☆☆Not started
Find the equivalent angle in the first revolution (between $0^\circ$ and $360^\circ$) for the angle of $1950^\circ$ and determine the sign of its secant and cotangent.
Find the equivalent angle in the first revolution (between $0^\circ$ and $360^\circ$) for the angle of $1950^\circ$ and determine the sign of its secant and cotangent.
Applications: gradients, angles of elevation and simplifying expressions
Trigonometric ratios describe any inclination: the height of a building from its shadow, the tilt of a solar panel, or an algebraic expression that is simplified using the Pythagorean identity.
Practice★★☆Not started
A building casts a shadow of $24$ metres when the angle of elevation of the sun above the horizon is $53^\circ$. Write the trigonometric expression to calculate the…
A building casts a shadow of $24$ metres when the angle of elevation of the sun above the horizon is $53^\circ$. Write the trigonometric expression to calculate the height of the building and determine its value. If the angle of elevation is halved, does the length of the shadow double? Justify your answer based on the non-linearity of the tangent ratio.
Simplify the following trigonometric expression as much as possible using fundamental identities, expressing the result in terms of a single trigonometric ratio:…
Simplify the following trigonometric expression as much as possible using fundamental identities, expressing the result in terms of a single trigonometric ratio:
Prove the following trigonometric identity, showing the algebraic steps and fundamental relations used. Assume that $\alpha$ is an angle for which all ratios are…
Prove the following trigonometric identity, showing the algebraic steps and fundamental relations used. Assume that $\alpha$ is an angle for which all ratios are defined: