Solving basic trigonometry value using quadrant rule or sum formula

Question

Basicly, i am bad at mathematics very hard to me got score 70. I try learn trigonometry now, starting from the very basic. I practice to find the value of $$\sin(135^\circ)$$ using my knowing about $\sin(45^\circ)={\sqrt2\over2}$, $\cos(45^\circ)={\sqrt2 \over 2}$, $\sin(90^\circ)=1$, $\cos(90^\circ)=0$, the quadrant rule and sum angle formula. I hope the expert can help me to correcting and giving a tips.

Using the quadrant rule : Since $135^\circ$ on second quadrant the sine sign is positive

Using sum formula of sin : $$\sin(135^\circ)=\sin(90^\circ+45^\circ)$$ $$=\sin(90^\circ)×\cos(45^\circ)+\cos(90^\circ)×\sin(45^\circ)$$ $$=1×\cos(45^\circ)+0×\sin(45^\circ)$$ $$=\cos(45^\circ)+0$$ $$=\cos(45^\circ)$$ $$={\sqrt2\over2}$$ For your informatiom, i am just economic faculty at second semester, and my college tutor giving us a calculus, that contain trigonometry. I need to learn this because 2 weeks again i got calculus examination. Thank you for your support and tips and explanation.

Is my practice is correct ? Can i get the simple way to remember the sum formula of sine and cosine ? What suitable practice for me after i practicing with $135^\circ$ ?

Answer 1

You don't need in that case use these formula, indeed by the definition of trigonometric function by the unit circle and by symmetry we can directly conclude that:

$$\sin(135^\circ)=\sin(90^\circ+45^\circ)=\sin 45°$$

Refer to the following image (credits):

Refer also to the related:

• What are some must-know trig identities?
• How to interpret trig functions in other quadrants?
• Coordinates of a point on a circle - review

Answer 2

$135^{\circ}$ is related to $45^{\circ}$ because $135^{\circ} = 180^{\circ} -45^{\circ}$ and as such, produces the positive or negative of the ratios concerning $45^{\circ}$. You don't need to know the compound angle formula to know this.